### Mixed Effects, hierarchical linear model (e.g., students nested in schools)
### This script shows both the lme4 and nlme package approaches to doing these models
### First, set the working directory where R will find the data
setwd("H:/data/User/Classes/multiv_13/YTs/YTSplit/Week 06 wmv - 2016/Wk06.R")
getwd()
## [1] "H:/data/User/Classes/multiv_13/YTs/YTSplit/Week 06 wmv - 2016/Wk06.R"
### Read the data
class06<- read.csv("Class06_homework_2014.csv")
#The lme4 package (and function lmer) are the more modern way of doing MLM in R
#At bottom, we reference a second package and function (nlme package, lme function), since many
#well-cited papers use this older (but more limited) approach
#Both methods are designed by Doug Bates and team
#install.packages("lme4")
library(lme4)
## Loading required package: Matrix
#This citation function is a cool tool to use with all packages
#Your manuscripts need to CITE the packages, and the "citation" function gives you that citation
citation("lme4")
##
## To cite lme4 in publications use:
##
## Douglas Bates, Martin Maechler, Ben Bolker, Steve Walker (2015).
## Fitting Linear Mixed-Effects Models Using lme4. Journal of
## Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01.
##
## A BibTeX entry for LaTeX users is
##
## @Article{,
## title = {Fitting Linear Mixed-Effects Models Using {lme4}},
## author = {Douglas Bates and Martin M{\"a}chler and Ben Bolker and Steve Walker},
## journal = {Journal of Statistical Software},
## year = {2015},
## volume = {67},
## number = {1},
## pages = {1--48},
## doi = {10.18637/jss.v067.i01},
## }
head(class06)
## ï..const ID BowlingTeamNo GroupGripStrength PercentWinBowl
## 1 1 1 1 65 51
## 2 1 2 1 65 51
## 3 1 3 1 65 46
## 4 1 4 1 65 50
## 5 1 5 1 65 62
## 6 1 6 1 65 45
## ForearmLengthQuartile WinningOrientation
## 1 2 52
## 2 2 43
## 3 1 42
## 4 1 44
## 5 4 35
## 6 1 49
### 0. Unconditional Model
#As we have done in other modeling types, the null model regresses your DV on an intercept (~1)
# This gives you the FIXED intercept
# The material in parentheses (1|school) gives you the random intercept -- i.e., each school's own intercept
# Also know as unconditional means model
nullmodel = lmer(WinningOrientation ~ 1 +
(1|BowlingTeamNo), data=class06, REML=FALSE)
summary(nullmodel)
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: WinningOrientation ~ 1 + (1 | BowlingTeamNo)
## Data: class06
##
## AIC BIC logLik deviance df.resid
## 16532.7 16550.1 -8263.4 16526.7 2428
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.23425 -0.78094 -0.02999 0.78812 2.49794
##
## Random effects:
## Groups Name Variance Std.Dev.
## BowlingTeamNo (Intercept) 12.84 3.583
## Residual 48.28 6.948
## Number of obs: 2431, groups: BowlingTeamNo, 102
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.550 0.382 124.5
#install.packages("stargazer")
library(stargazer)
##
## Please cite as:
## Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.
## R package version 5.2.2. https://CRAN.R-project.org/package=stargazer
stargazer(nullmodel, type = "text",
digits = 3,
star.cutoffs = c(0.05, 0.01, 0.001),
digit.separator = "")
##
## =================================================
## Dependent variable:
## -----------------------------
## WinningOrientation
## -------------------------------------------------
## Constant 47.550***
## (0.382)
##
## -------------------------------------------------
## Observations 2431
## Log Likelihood -8263.358
## Akaike Inf. Crit. 16532.720
## Bayesian Inf. Crit. 16550.100
## =================================================
## Note: *p<0.05; **p<0.01; ***p<0.001
#install.packages("sjPlot")
library(sjPlot)
tab_model(nullmodel)
|
|
WinningOrientation
|
|
Predictors
|
Estimates
|
CI
|
p
|
|
(Intercept)
|
47.55
|
46.80 – 48.30
|
<0.001
|
|
Random Effects
|
|
σ2
|
48.28
|
|
Ï„00 BowlingTeamNo
|
12.84
|
|
ICC
|
0.21
|
|
N BowlingTeamNo
|
102
|
|
Observations
|
2431
|
|
Marginal R2 / Conditional R2
|
0.000 / 0.210
|
#Saving predicted values to the data frame, so we can plot what the null model looks like
pred_0<-(fitted(nullmodel))
pred0<-as.data.frame(pred_0)
class06<-cbind(class06,pred0)
library(ggplot2)
#This plot will show random intercept, with no predictors
qplot(y=pred_0, x=PercentWinBowl, colour = BowlingTeamNo,
data = class06)
#This plot will superimpose the fixed intercept and null slope, just to show that this
#first model is a nothing
ggplot(class06) +
aes(x=class06$PercentWinBowl, y=class06$pred_0) + #, col=as.factor(hsb_pred2$school)
geom_point() +
stat_smooth(method = "lm", se = FALSE)
## Warning: Use of `class06$PercentWinBowl` is discouraged. Use `PercentWinBowl`
## instead.
## Warning: Use of `class06$pred_0` is discouraged. Use `pred_0` instead.
## Warning: Use of `class06$PercentWinBowl` is discouraged. Use `PercentWinBowl`
## instead.
## Warning: Use of `class06$pred_0` is discouraged. Use `pred_0` instead.
## `geom_smooth()` using formula 'y ~ x'
### 1. Conditional model: Fixed effect of Level 2 GroupGripStrength
condmodel1 = lmer(WinningOrientation ~ GroupGripStrength +
(1|BowlingTeamNo), data=class06, REML=FALSE)
summary(condmodel1)
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: WinningOrientation ~ GroupGripStrength + (1 | BowlingTeamNo)
## Data: class06
##
## AIC BIC logLik deviance df.resid
## 16534.7 16557.9 -8263.3 16526.7 2427
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.23668 -0.77876 -0.02977 0.78740 2.49476
##
## Random effects:
## Groups Name Variance Std.Dev.
## BowlingTeamNo (Intercept) 12.83 3.582
## Residual 48.28 6.948
## Number of obs: 2431, groups: BowlingTeamNo, 102
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.971132 1.948607 24.62
## GroupGripStrength -0.008428 0.038233 -0.22
##
## Correlation of Fixed Effects:
## (Intr)
## GrpGrpStrng -0.981
#lower case anova does nested model tests
anova(nullmodel,condmodel1)
## Data: class06
## Models:
## nullmodel: WinningOrientation ~ 1 + (1 | BowlingTeamNo)
## condmodel1: WinningOrientation ~ GroupGripStrength + (1 | BowlingTeamNo)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## nullmodel 3 16533 16550 -8263.4 16527
## condmodel1 4 16535 16558 -8263.3 16527 0.0486 1 0.8256
#Saving predicted values to the data frame, so we can plot what the conditional model looks like
pred_1<-(fitted(condmodel1))
pred1<-as.data.frame(pred_1)
class06<-cbind(class06,pred1)
#This plot will show the overall fixed intercept and slope; a Level 2 factor cannot
#have random variance
qplot(y=pred_1, x=GroupGripStrength, colour = BowlingTeamNo,
data = class06)
#The second plot imposes the best fitting line
ggplot(class06) +
aes(x=class06$GroupGripStrength, y=class06$pred_1) + #, col=as.factor(hsb_pred2$school)
geom_point() +
stat_smooth(method = "lm", se = FALSE)
## Warning: Use of `class06$GroupGripStrength` is discouraged. Use
## `GroupGripStrength` instead.
## Warning: Use of `class06$pred_1` is discouraged. Use `pred_1` instead.
## Warning: Use of `class06$GroupGripStrength` is discouraged. Use
## `GroupGripStrength` instead.
## Warning: Use of `class06$pred_1` is discouraged. Use `pred_1` instead.
## `geom_smooth()` using formula 'y ~ x'
#Gives you each school's fixed and random! coefficients
coef(condmodel1)
## $BowlingTeamNo
## (Intercept) GroupGripStrength
## 1 44.36353 -0.008427987
## 2 43.72894 -0.008427987
## 3 43.72935 -0.008427987
## 4 42.35377 -0.008427987
## 5 43.49744 -0.008427987
## 6 43.92238 -0.008427987
## 7 41.57389 -0.008427987
## 8 43.03649 -0.008427987
## 9 43.39790 -0.008427987
## 10 43.23115 -0.008427987
## 11 44.59257 -0.008427987
## 12 46.43803 -0.008427987
## 13 42.95839 -0.008427987
## 14 42.35983 -0.008427987
## 15 44.91674 -0.008427987
## 16 45.34331 -0.008427987
## 17 45.27734 -0.008427987
## 18 45.20489 -0.008427987
## 19 45.37000 -0.008427987
## 20 45.44204 -0.008427987
## 21 42.95880 -0.008427987
## 22 46.95564 -0.008427987
## 23 44.77995 -0.008427987
## 24 42.34485 -0.008427987
## 25 43.69170 -0.008427987
## 26 44.37687 -0.008427987
## 27 47.38139 -0.008427987
## 28 42.97820 -0.008427987
## 29 43.90012 -0.008427987
## 30 46.48862 -0.008427987
## 31 47.77555 -0.008427987
## 32 46.75451 -0.008427987
## 33 45.70350 -0.008427987
## 34 45.49427 -0.008427987
## 35 42.92720 -0.008427987
## 36 47.76303 -0.008427987
## 37 46.97750 -0.008427987
## 38 43.78682 -0.008427987
## 39 44.97339 -0.008427987
## 40 46.37328 -0.008427987
## 41 45.67960 -0.008427987
## 42 48.30331 -0.008427987
## 43 47.97709 -0.008427987
## 44 47.33645 -0.008427987
## 45 46.48093 -0.008427987
## 46 46.97586 -0.008427987
## 47 49.82861 -0.008427987
## 48 48.28104 -0.008427987
## 49 49.83590 -0.008427987
## 50 48.92938 -0.008427987
## 51 47.20695 -0.008427987
## 52 46.87673 -0.008427987
## 53 47.63312 -0.008427987
## 54 47.37288 -0.008427987
## 55 47.06975 -0.008427987
## 56 48.98602 -0.008427987
## 57 50.03908 -0.008427987
## 58 46.42911 -0.008427987
## 59 48.92250 -0.008427987
## 60 48.02203 -0.008427987
## 61 49.15236 -0.008427987
## 62 49.14590 -0.008427987
## 63 51.01240 -0.008427987
## 64 49.80962 -0.008427987
## 65 50.39844 -0.008427987
## 66 48.39032 -0.008427987
## 67 50.06134 -0.008427987
## 68 49.82174 -0.008427987
## 69 50.69347 -0.008427987
## 70 51.83960 -0.008427987
## 71 49.51255 -0.008427987
## 72 50.98244 -0.008427987
## 73 49.55585 -0.008427987
## 74 48.30977 -0.008427987
## 75 50.80922 -0.008427987
## 76 50.11030 -0.008427987
## 77 49.65057 -0.008427987
## 78 52.14232 -0.008427987
## 79 49.88813 -0.008427987
## 80 51.21435 -0.008427987
## 81 51.28638 -0.008427987
## 82 50.83836 -0.008427987
## 83 51.85540 -0.008427987
## 84 50.59311 -0.008427987
## 85 52.23662 -0.008427987
## 86 51.71779 -0.008427987
## 87 51.91769 -0.008427987
## 88 52.15525 -0.008427987
## 89 49.92333 -0.008427987
## 90 49.73554 -0.008427987
## 91 49.33409 -0.008427987
## 92 51.35678 -0.008427987
## 93 52.03426 -0.008427987
## 94 52.50168 -0.008427987
## 95 54.03067 -0.008427987
## 96 51.76715 -0.008427987
## 97 54.64217 -0.008427987
## 98 53.70651 -0.008427987
## 99 53.50333 -0.008427987
## 100 53.57537 -0.008427987
## 101 53.38922 -0.008427987
## 102 55.14457 -0.008427987
##
## attr(,"class")
## [1] "coef.mer"
library(car)
## Loading required package: carData
## Registered S3 methods overwritten by 'car':
## method from
## influence.merMod lme4
## cooks.distance.influence.merMod lme4
## dfbeta.influence.merMod lme4
## dfbetas.influence.merMod lme4
#Gives you chi-square for each model term
Anova(condmodel1)
## Analysis of Deviance Table (Type II Wald chisquare tests)
##
## Response: WinningOrientation
## Chisq Df Pr(>Chisq)
## GroupGripStrength 0.0486 1 0.8255
#Three different approaches for confidence intervals
#Method wald is an approximation (same as spss), but fast
#the first and third methods (likelihood and bootstrapping) take a long time
confint(condmodel1) #Profile method; takes a while, less stringent assumptions
## Computing profile confidence intervals ...
## 2.5 % 97.5 %
## .sig01 3.05971909 4.22333606
## .sigma 6.75326002 7.15269729
## (Intercept) 44.11520888 51.82780845
## GroupGripStrength -0.08408385 0.06724368
confint(condmodel1, method = "Wald")
## 2.5 % 97.5 %
## .sig01 NA NA
## .sigma NA NA
## (Intercept) 44.15193200 51.79033249
## GroupGripStrength -0.08336383 0.06650786
confint(condmodel1, method = "boot",
nsim = 1000,
boot.type = "perc", #,"basic","norm" are options
FUN = NULL, quiet = FALSE)
## Computing bootstrap confidence intervals ...
## 2.5 % 97.5 %
## .sig01 2.96017455 4.0954240
## .sigma 6.74975811 7.1505102
## (Intercept) 43.95822925 51.8410759
## GroupGripStrength -0.08391973 0.0725048
#This ensures that your model is stored in a lmerMod object, needed for the next step
class(condmodel1) <- "lmerMod"
#Stargazer makes a nicely formated model summary table (albeit incomplete)
stargazer(condmodel1, type = "text",
digits = 3,
star.cutoffs = c(0.05, 0.01, 0.001),
digit.separator = "")
##
## =================================================
## Dependent variable:
## -----------------------------
## WinningOrientation
## -------------------------------------------------
## GroupGripStrength -0.008
## (0.038)
##
## Constant 47.971***
## (1.949)
##
## -------------------------------------------------
## Observations 2431
## Log Likelihood -8263.334
## Akaike Inf. Crit. 16534.670
## Bayesian Inf. Crit. 16557.850
## =================================================
## Note: *p<0.05; **p<0.01; ***p<0.001
###sjPlot::tab_model is even better, because it formats the table in APA style and adds
# fixed and random effects and p-values
tab_model(nullmodel,condmodel1)
|
|
WinningOrientation
|
WinningOrientation
|
|
Predictors
|
Estimates
|
CI
|
p
|
Estimates
|
CI
|
p
|
|
(Intercept)
|
47.55
|
46.80 – 48.30
|
<0.001
|
47.97
|
44.15 – 51.79
|
<0.001
|
|
GroupGripStrength
|
|
|
|
-0.01
|
-0.08 – 0.07
|
0.826
|
|
Random Effects
|
|
σ2
|
48.28
|
48.28
|
|
Ï„00
|
12.84 BowlingTeamNo
|
12.83 BowlingTeamNo
|
|
ICC
|
0.21
|
0.21
|
|
N
|
102 BowlingTeamNo
|
102 BowlingTeamNo
|
|
Observations
|
2431
|
2431
|
|
Marginal R2 / Conditional R2
|
0.000 / 0.210
|
0.000 / 0.210
|
#Exact p-values and df in revised syntax below, but warning VERY time consuming (crashed my computer)
#tab_model(condmodel2, p.val = "kr", show.df = TRUE)
# plot to evaluate normality of residuals
qqnorm(resid(condmodel1))
qqline(resid(condmodel1)) # points fall nicely onto the line - good!
#plots to evaluate linearity and homoscedasticity of residuals
plot(condmodel1, type = c("p", "smooth"))
plot(condmodel1, sqrt(abs(resid(.))) ~ fitted(.),
type = c("p", "smooth"),col="black")
# There is much additional output you can request from the lmer model object
extractAIC(condmodel1) #model AIC only
## [1] 4.00 16534.67
fixef(condmodel1) #model fixed effects only
## (Intercept) GroupGripStrength
## 47.971132244 -0.008427987
formula(condmodel1) #confirmation of what formula the model estimated
## WinningOrientation ~ GroupGripStrength + (1 | BowlingTeamNo)
nobs(condmodel1) #number of Level 1 observations
## [1] 2431
print(condmodel1) #alternative to summary()
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: WinningOrientation ~ GroupGripStrength + (1 | BowlingTeamNo)
## Data: class06
## AIC BIC logLik deviance df.resid
## 16534.668 16557.852 -8263.334 16526.668 2427
## Random effects:
## Groups Name Std.Dev.
## BowlingTeamNo (Intercept) 3.582
## Residual 6.948
## Number of obs: 2431, groups: BowlingTeamNo, 102
## Fixed Effects:
## (Intercept) GroupGripStrength
## 47.971132 -0.008428
ranef(condmodel1) #random effects; shows each school's deviation from fixed intercept and cses
## $BowlingTeamNo
## (Intercept)
## 1 -3.607605401
## 2 -4.242187329
## 3 -4.241778215
## 4 -5.617361411
## 5 -4.473688238
## 6 -4.048750591
## 7 -6.397245078
## 8 -4.934644484
## 9 -4.573231150
## 10 -4.739980403
## 11 -3.378559178
## 12 -1.533097736
## 13 -5.012739882
## 14 -5.611303211
## 15 -3.054391786
## 16 -2.627817682
## 17 -2.693796680
## 18 -2.766242993
## 19 -2.601130197
## 20 -2.529092998
## 21 -5.012330768
## 22 -1.015493605
## 23 -3.191180640
## 24 -5.626283411
## 25 -4.279433271
## 26 -3.594261659
## 27 -0.589737730
## 28 -4.992928826
## 29 -4.071016333
## 30 -1.482508052
## 31 -0.195578711
## 32 -1.216625001
## 33 -2.267631690
## 34 -2.476866857
## 35 -5.043927624
## 36 -0.208104225
## 37 -0.993636977
## 38 -4.184312102
## 39 -2.997743901
## 40 -1.597849392
## 41 -2.291533890
## 42 0.332174763
## 43 0.005961799
## 44 -0.634678329
## 45 -1.490202709
## 46 -0.995273434
## 47 1.857481441
## 48 0.309909020
## 49 1.864766984
## 50 0.958243805
## 51 -0.764181640
## 52 -1.094407232
## 53 -0.338016650
## 54 -0.598250615
## 55 -0.901379608
## 56 1.014891689
## 57 2.067943950
## 58 -1.542019736
## 59 0.951367377
## 60 0.050902398
## 61 1.181231828
## 62 1.174764514
## 63 3.041264356
## 64 1.838488613
## 65 2.427311713
## 66 0.419192161
## 67 2.090209692
## 68 1.850605013
## 69 2.722336935
## 70 3.868464793
## 71 1.541417820
## 72 3.011303956
## 73 1.584721962
## 74 0.338642077
## 75 2.838087389
## 76 2.139162920
## 77 1.679434016
## 78 4.171184672
## 79 1.916993125
## 80 3.243213979
## 81 3.315251178
## 82 2.867229560
## 83 3.884263221
## 84 2.621975794
## 85 4.265487612
## 86 3.746656139
## 87 3.946560192
## 88 4.184119300
## 89 1.952193496
## 90 1.764405843
## 91 1.362961281
## 92 3.385651919
## 93 4.063128874
## 94 4.530552435
## 95 6.059541141
## 96 3.796018481
## 97 6.671039099
## 98 5.735373749
## 99 5.532196783
## 100 5.604233981
## 101 5.418082785
## 102 7.173441638
##
## with conditional variances for "BowlingTeamNo"
residuals(condmodel1) #provides each participant's residuals
## 1 2 3 4 5 6
## 8.18429230 -0.81570770 -1.81570770 0.18429230 -8.81570770 5.18429230
## 7 8 9 10 11 12
## 2.18429230 -0.81570770 6.18429230 0.18429230 4.18429230 -0.81570770
## 13 14 15 16 17 18
## -9.81570770 -13.81570770 3.18429230 4.18429230 -9.81570770 -7.81570770
## 19 20 21 22 23 24
## 2.18429230 0.18429230 8.18429230 -4.81570770 -2.81570770 4.18429230
## 25 26 27 28 29 30
## -9.24854967 -5.24854967 4.75145033 -3.24854967 -2.24854967 2.75145033
## 31 32 33 34 35 36
## 6.75145033 2.75145033 5.75145033 0.75145033 -1.24854967 -11.24854967
## 37 38 39 40 41 42
## -15.24854967 0.75145033 -10.24854967 6.75145033 -1.24854967 0.75145033
## 43 44 45 46 47 48
## -2.24854967 -5.24854967 -3.24854967 6.75145033 4.75145033 10.75145033
## 49 50 51 52 53 54
## -4.20681885 -0.20681885 1.79318115 4.79318115 -6.20681885 -10.20681885
## 55 56 57 58 59 60
## 7.79318115 5.79318115 2.79318115 8.79318115 -0.20681885 9.79318115
## 61 62 63 64 65 66
## -8.20681885 -12.20681885 8.79318115 -3.20681885 -7.20681885 -8.20681885
## 67 68 69 70 71 72
## 6.79318115 -8.20681885 2.79318115 -14.20681885 9.79318115 -3.20681885
## 73 74 75 76 77 78
## -6.79752370 -3.79752370 -3.79752370 4.20247630 -3.79752370 1.20247630
## 79 80 81 82 83 84
## 1.20247630 7.20247630 -10.79752370 4.20247630 -4.79752370 -6.79752370
## 85 86 87 88 89 90
## 1.20247630 -2.79752370 12.20247630 -5.79752370 -3.79752370 5.20247630
## 91 92 93 94 95 96
## -5.79752370 -9.79752370 11.20247630 2.20247630 -0.79752370 -1.79752370
## 97 98 99 100 101 102
## -7.11818460 -4.11818460 -0.11818460 3.88181540 3.88181540 -3.11818460
## 103 104 105 106 107 108
## 5.88181540 7.88181540 2.88181540 1.88181540 -9.11818460 13.88181540
## 109 110 111 112 113 114
## -7.11818460 4.88181540 -12.11818460 1.88181540 -5.11818460 -8.11818460
## 115 116 117 118 119 120
## -1.11818460 1.88181540 -6.11818460 7.88181540 -12.11818460 1.88181540
## 121 122 123 124 125 126
## -2.55155023 -4.55155023 -5.55155023 -3.55155023 -7.55155023 -0.55155023
## 127 128 129 130 131 132
## -1.55155023 12.44844977 10.44844977 8.44844977 -5.55155023 -9.55155023
## 133 134 135 136 137 138
## 2.44844977 0.44844977 -6.55155023 2.44844977 -9.55155023 2.44844977
## 139 140 141 142 143 144
## -9.55155023 -3.55155023 4.44844977 6.44844977 10.44844977 -5.55155023
## 145 146 147 148 149 150
## 9.78851627 0.78851627 -8.21148373 9.78851627 -3.21148373 4.78851627
## 151 152 153 154 155 156
## 4.78851627 3.78851627 -0.21148373 -2.21148373 -6.21148373 -2.21148373
## 157 158 159 160 161 162
## -11.21148373 -3.21148373 -7.21148373 -5.21148373 0.78851627 1.78851627
## 163 164 165 166 167 168
## 8.78851627 -4.21148373 8.78851627 -8.21148373 -9.21148373 -7.21148373
## 169 170 171 172 173 174
## -2.64880037 -9.64880037 3.35119963 9.35119963 -0.64880037 -10.64880037
## 175 176 177 178 179 180
## 6.35119963 -6.64880037 -4.64880037 -5.64880037 11.35119963 8.35119963
## 181 182 183 184 185 186
## -8.64880037 0.35119963 -6.64880037 11.35119963 3.35119963 -2.64880037
## 187 188 189 190 191 192
## -0.64880037 -8.64880037 12.35119963 -1.64880037 -2.64880037 -12.64880037
## 193 194 195 196 197 198
## -4.88379390 2.11620610 6.11620610 -9.88379390 -5.88379390 8.11620610
## 199 200 201 202 203 204
## 4.11620610 -10.88379390 4.11620610 -3.88379390 9.11620610 -5.88379390
## 205 206 207 208 209 210
## -0.88379390 -0.88379390 1.11620610 10.11620610 -11.88379390 -2.88379390
## 211 212 213 214 215 216
## -0.88379390 4.11620610 0.11620610 -9.88379390 -1.88379390 4.11620610
## 217 218 219 220 221 222
## 2.17339153 -0.82660847 9.17339153 -0.82660847 4.17339153 2.17339153
## 223 224 225 226 227 228
## -12.82660847 -1.82660847 -13.82660847 -13.82660847 -3.82660847 14.17339153
## 229 230 231 232 233 234
## 2.17339153 -8.82660847 8.17339153 0.17339153 -2.82660847 -4.82660847
## 235 236 237 238 239 240
## -1.82660847 10.17339153 7.17339153 -10.82660847 1.17339153 -1.82660847
## 241 242 243 244 245 246
## -8.19645769 -3.19645769 10.80354231 10.80354231 -1.19645769 -3.19645769
## 247 248 249 250 251 252
## 0.80354231 7.80354231 9.80354231 2.80354231 13.80354231 -2.19645769
## 253 254 255 256 257 258
## -3.19645769 8.80354231 6.80354231 -7.19645769 -0.19645769 -12.19645769
## 259 260 261 262 263 264
## -11.19645769 -0.19645769 -13.19645769 -12.19645769 -6.19645769 -1.19645769
## 265 266 267 268 269 270
## 6.09292866 2.09292866 11.09292866 -2.90707134 -7.90707134 8.09292866
## 271 272 273 274 275 276
## -1.90707134 3.09292866 -7.90707134 7.09292866 -8.90707134 -0.90707134
## 277 278 279 280 281 282
## 7.09292866 8.09292866 2.09292866 7.09292866 1.09292866 -1.90707134
## 283 284 285 286 287 288
## -9.90707134 -5.90707134 -0.90707134 4.09292866 -13.90707134 -9.90707134
## 289 290 291 292 293 294
## 11.54728685 -9.45271315 -0.45271315 -1.45271315 -8.45271315 2.54728685
## 295 296 297 298 299 300
## 4.54728685 -4.45271315 -8.45271315 5.54728685 5.54728685 6.54728685
## 301 302 303 304 305 306
## -7.45271315 -4.45271315 -0.45271315 -6.45271315 -8.45271315 -8.45271315
## 307 308 309 310 311 312
## 1.54728685 6.54728685 -2.45271315 -6.45271315 9.54728685 4.54728685
## 313 314 315 316 317 318
## -5.92157372 2.07842628 9.07842628 -5.92157372 -3.92157372 7.07842628
## 319 320 321 322 323 324
## -8.92157372 -7.92157372 4.07842628 -3.92157372 -7.92157372 -5.92157372
## 325 326 327 328 329 330
## 0.07842628 4.07842628 -10.92157372 5.07842628 -12.92157372 13.07842628
## 331 332 333 334 335 336
## 1.07842628 0.07842628 -5.92157372 9.07842628 0.07842628 4.07842628
## 337 338 339 340 341 342
## 10.47937492 -8.52062508 -4.52062508 -3.52062508 5.47937492 -1.52062508
## 343 344 345 346 347 348
## 4.47937492 -5.52062508 10.47937492 -2.52062508 2.47937492 3.47937492
## 349 350 351 352 353 354
## -1.52062508 2.47937492 -8.52062508 0.47937492 -3.52062508 5.47937492
## 355 356 357 358 359 360
## -4.52062508 -7.52062508 5.47937492 -6.52062508 -7.52062508 3.47937492
## 361 362 363 364 365 366
## 10.21293257 12.21293257 -11.78706743 -3.78706743 4.21293257 4.21293257
## 367 368 369 370 371 372
## 8.21293257 -3.78706743 -8.78706743 -9.78706743 0.21293257 0.21293257
## 373 374 375 376 377 378
## -1.78706743 0.21293257 1.21293257 -3.78706743 -0.78706743 -8.78706743
## 379 380 381 382 383 384
## -7.78706743 3.21293257 2.21293257 -11.78706743 4.21293257 12.21293257
## 385 386 387 388 389 390
## -6.83908025 2.16091975 1.16091975 2.16091975 -9.83908025 0.16091975
## 391 392 393 394 395 396
## -2.83908025 -11.83908025 -8.83908025 -2.83908025 -1.83908025 8.16091975
## 397 398 399 400 401 402
## -9.83908025 4.16091975 -3.83908025 5.16091975 -3.83908025 8.16091975
## 403 404 405 406 407 408
## 8.16091975 3.16091975 13.16091975 -9.83908025 -1.83908025 8.16091975
## 409 410 411 412 413 414
## -4.80877387 -1.80877387 -1.80877387 -10.80877387 1.19122613 1.19122613
## 415 416 417 418 419 420
## -4.80877387 1.19122613 1.19122613 9.19122613 -2.80877387 8.19122613
## 421 422 423 424 425 426
## -14.80877387 -9.80877387 3.19122613 4.19122613 -6.80877387 3.19122613
## 427 428 429 430 431 432
## -4.80877387 8.19122613 10.19122613 7.19122613 -1.80877387 -3.80877387
## 433 434 435 436 437 438
## 0.96711742 2.96711742 8.96711742 -10.03288258 -5.03288258 6.96711742
## 439 440 441 442 443 444
## -5.03288258 -3.03288258 0.96711742 -8.03288258 3.96711742 5.96711742
## 445 446 447 448 449 450
## 0.96711742 -0.03288258 4.96711742 6.96711742 -1.03288258 6.96711742
## 451 452 453 454 455 456
## -3.03288258 -5.03288258 -11.03288258 -3.03288258 -1.03288258 -5.03288258
## 457 458 459 460 461 462
## 4.89508023 0.89508023 -5.10491977 -13.10491977 -5.10491977 -8.10491977
## 463 464 465 466 467 468
## 0.89508023 -7.10491977 5.89508023 -3.10491977 5.89508023 -14.10491977
## 469 470 471 472 473 474
## 0.89508023 10.89508023 -15.10491977 10.89508023 3.89508023 -8.10491977
## 475 476 477 478 479 480
## -1.10491977 9.89508023 6.89508023 -3.10491977 7.89508023 3.89508023
## 481 482 483 484 485 486
## -8.41098233 -5.41098233 -4.41098233 5.58901767 -5.41098233 10.58901767
## 487 488 489 490 491 492
## -4.41098233 4.58901767 -6.41098233 -9.41098233 0.58901767 -4.41098233
## 493 494 495 496 497 498
## -5.41098233 -1.41098233 -3.41098233 -3.41098233 -6.41098233 12.58901767
## 499 500 501 502 503 504
## 4.58901767 3.58901767 4.58901767 -1.41098233 0.58901767 3.58901767
## 505 506 507 508 509 510
## 3.46576070 0.46576070 -5.53423930 3.46576070 6.46576070 2.46576070
## 511 512 513 514 515 516
## 10.46576070 8.46576070 -3.53423930 6.46576070 9.46576070 2.46576070
## 517 518 519 520 521 522
## -11.53423930 -11.53423930 -2.53423930 5.46576070 4.46576070 -5.53423930
## 523 524 525 526 527 528
## 8.46576070 -14.53423930 -7.53423930 1.46576070 -12.53423930 -2.53423930
## 529 530 531 532 533 534
## 8.62459176 -6.37540824 8.62459176 -5.37540824 -5.37540824 2.62459176
## 535 536 537 538 539 540
## -2.37540824 5.62459176 2.62459176 -2.37540824 -1.37540824 -9.37540824
## 541 542 543 544 545 546
## -3.37540824 -5.37540824 -8.37540824 -4.37540824 6.62459176 -6.37540824
## 547 548 549 550 551 552
## 10.62459176 0.62459176 4.62459176 -3.37540824 -0.37540824 1.62459176
## 553 554 555 556 557 558
## -0.96558943 -3.96558943 -8.96558943 -1.96558943 -5.96558943 -5.96558943
## 559 560 561 562 563 564
## -1.96558943 2.03441057 -2.96558943 0.03441057 -0.96558943 -5.96558943
## 565 566 567 568 569 570
## 1.03441057 2.03441057 8.03441057 -3.96558943 15.03441057 0.03441057
## 571 572 573 574 575 576
## -3.96558943 8.03441057 -9.96558943 14.03441057 -3.96558943 -9.96558943
## 577 578 579 580 581 582
## 3.66227647 -7.33772353 0.66227647 -6.33772353 4.66227647 -12.33772353
## 583 584 585 586 587 588
## 13.66227647 8.66227647 10.66227647 -5.33772353 -3.33772353 -1.33772353
## 589 590 591 592 593 594
## -7.33772353 -7.33772353 -6.33772353 -0.33772353 3.66227647 -5.33772353
## 595 596 597 598 599 600
## 1.66227647 6.66227647 8.66227647 -5.33772353 2.66227647 -13.33772353
## 601 602 603 604 605 606
## -7.93861527 -5.93861527 4.06138473 -1.93861527 3.06138473 -5.93861527
## 607 608 609 610 611 612
## 2.06138473 0.06138473 8.06138473 8.06138473 -1.93861527 -4.93861527
## 613 614 615 616 617 618
## 1.06138473 -2.93861527 2.06138473 -11.93861527 12.06138473 5.06138473
## 619 620 621 622 623 624
## 0.06138473 -7.93861527 8.06138473 -2.93861527 -1.93861527 -10.93861527
## 625 626 627 628 629 630
## -8.88414329 -1.88414329 -7.88414329 -4.88414329 1.11585671 0.11585671
## 631 632 633 634 635 636
## -8.88414329 7.11585671 2.11585671 2.11585671 -9.88414329 6.11585671
## 637 638 639 640 641 642
## -9.88414329 2.11585671 8.11585671 11.11585671 1.11585671 4.11585671
## 643 644 645 646 647 648
## 1.11585671 0.11585671 0.11585671 6.11585671 6.11585671 -8.88414329
## 649 650 651 652 653 654
## 6.34206008 -2.65793992 -7.65793992 12.34206008 -9.65793992 -9.65793992
## 655 656 657 658 659 660
## 6.34206008 -1.65793992 14.34206008 0.34206008 2.34206008 -3.65793992
## 661 662 663 664 665 666
## -3.65793992 -10.65793992 -5.65793992 -3.65793992 -2.65793992 6.34206008
## 667 668 669 670 671 672
## -7.65793992 -3.65793992 -9.65793992 10.34206008 -7.65793992 12.34206008
## 673 674 675 676 677 678
## -8.59670839 -7.59670839 0.40329161 13.40329161 7.40329161 -12.59670839
## 679 680 681 682 683 684
## 3.40329161 -3.59670839 8.40329161 9.40329161 -4.59670839 1.40329161
## 685 686 687 688 689 690
## -9.59670839 -5.59670839 -10.59670839 -5.59670839 7.40329161 8.40329161
## 691 692 693 694 695 696
## -6.59670839 -6.59670839 3.40329161 -2.59670839 -4.59670839 10.40329161
## 697 698 699 700 701 702
## 3.05919495 -13.94080505 6.05919495 -2.94080505 -6.94080505 6.05919495
## 703 704 705 706 707 708
## -3.94080505 -9.94080505 -2.94080505 -4.94080505 7.05919495 3.05919495
## 709 710 711 712 713 714
## 0.05919495 0.05919495 -7.94080505 -1.94080505 8.05919495 -3.94080505
## 715 716 717 718 719 720
## 4.05919495 -3.94080505 -5.94080505 8.05919495 8.05919495 10.05919495
## 721 722 723 724 725 726
## -1.48900198 6.51099802 -9.48900198 0.51099802 -1.48900198 6.51099802
## 727 728 729 730 731 732
## -11.48900198 9.51099802 8.51099802 -1.48900198 -7.48900198 -0.48900198
## 733 734 735 736 737 738
## 4.51099802 4.51099802 -11.48900198 2.51099802 4.51099802 -2.48900198
## 739 740 741 742 743 744
## -7.48900198 -5.48900198 0.51099802 8.51099802 2.51099802 0.51099802
## 745 746 747 748 749 750
## 3.72588800 5.72588800 4.72588800 -0.27411200 -6.27411200 -11.27411200
## 751 752 753 754 755 756
## 8.72588800 7.72588800 -10.27411200 5.72588800 0.72588800 -13.27411200
## 757 758 759 760 761 762
## 0.72588800 7.72588800 -4.27411200 -5.27411200 7.72588800 5.72588800
## 763 764 765 766 767 768
## 5.72588800 -3.27411200 2.72588800 -5.27411200 -2.27411200 -10.27411200
## 769 770 771 772 773 774
## -4.14725342 -2.14725342 7.85274658 -10.14725342 -0.14725342 -5.14725342
## 775 776 777 778 779 780
## 7.85274658 5.85274658 9.85274658 5.85274658 1.85274658 -12.14725342
## 781 782 783 784 785 786
## 11.85274658 11.85274658 -2.14725342 -5.14725342 -12.14725342 3.85274658
## 787 788 789 790 791 792
## 1.85274658 -12.14725342 -1.14725342 -7.14725342 -8.14725342 4.85274658
## 793 794 795 796 797 798
## -3.97173021 9.02826979 -6.97173021 -4.97173021 -3.97173021 -1.97173021
## 799 800 801 802 803 804
## -3.97173021 0.02826979 -0.97173021 5.02826979 -3.97173021 4.02826979
## 805 806 807 808 809 810
## 1.02826979 -2.97173021 4.02826979 -5.97173021 -2.97173021 -1.97173021
## 811 812 813 814 815 816
## -8.97173021 3.02826979 -4.97173021 4.02826979 7.02826979 12.02826979
## 817 818 819 820 821 822
## -5.66593703 8.33406297 -5.66593703 -3.66593703 -3.66593703 -0.66593703
## 823 824 825 826 827 828
## -7.66593703 -5.66593703 9.33406297 -5.66593703 6.33406297 2.33406297
## 829 830 831 832 833 834
## -7.66593703 -7.66593703 -7.66593703 6.33406297 -10.66593703 17.33406297
## 835 836 837 838 839 840
## 8.33406297 -1.66593703 10.33406297 -6.66593703 -7.66593703 0.33406297
## 841 842 843 844 845 846
## -0.28263277 6.71736723 10.71736723 4.71736723 -15.28263277 -11.28263277
## 847 848 849 850 851 852
## -8.28263277 -5.28263277 -8.28263277 -3.28263277 2.71736723 -2.28263277
## 853 854 855 856 857 858
## 8.71736723 8.71736723 10.71736723 -2.28263277 -10.28263277 8.71736723
## 859 860 861 862 863 864
## -1.28263277 10.71736723 -9.28263277 -5.28263277 4.71736723 4.71736723
## 865 866 867 868 869 870
## -4.53081197 -6.53081197 -12.53081197 13.46918803 -1.53081197 8.46918803
## 871 872 873 874 875 876
## 10.46918803 7.46918803 1.46918803 -8.53081197 3.46918803 -5.53081197
## 877 878 879 880 881 882
## -2.53081197 -8.53081197 1.46918803 -9.53081197 9.46918803 -10.53081197
## 883 884 885 886 887 888
## -5.53081197 5.46918803 12.46918803 -6.53081197 5.46918803 -0.53081197
## 889 890 891 892 893 894
## -8.28114093 3.71885907 10.71885907 0.71885907 -9.28114093 -5.28114093
## 895 896 897 898 899 900
## -0.28114093 -4.28114093 -3.28114093 9.71885907 -2.28114093 4.71885907
## 901 902 903 904 905 906
## -6.28114093 -10.28114093 -8.28114093 9.71885907 -4.28114093 4.71885907
## 907 908 909 910 911 912
## -6.28114093 0.71885907 13.71885907 1.71885907 -3.28114093 -4.28114093
## 913 914 915 916 917 918
## -3.67840880 4.32159120 -9.67840880 -1.67840880 0.32159120 12.32159120
## 919 920 921 922 923 924
## -4.67840880 5.32159120 10.32159120 -7.67840880 -5.67840880 -3.67840880
## 925 926 927 928 929 930
## -1.67840880 -0.67840880 12.32159120 3.32159120 -3.67840880 -1.67840880
## 931 932 933 934 935 936
## 4.32159120 -3.67840880 -7.67840880 -7.67840880 0.32159120 -0.67840880
## 937 938 939 940 941 942
## 2.16610830 11.16610830 -2.83389170 4.16610830 0.16610830 -0.83389170
## 943 944 945 946 947 948
## 7.16610830 5.16610830 9.16610830 -11.83389170 1.16610830 -8.83389170
## 949 950 951 952 953 954
## 9.16610830 -7.83389170 1.16610830 -4.83389170 -4.83389170 1.16610830
## 955 956 957 958 959 960
## -0.83389170 -2.83389170 9.16610830 -10.83389170 -12.83389170 2.16610830
## 961 962 963 964 965 966
## 3.64066514 -7.35933486 0.64066514 -8.35933486 -0.35933486 10.64066514
## 967 968 969 970 971 972
## -1.35933486 -3.35933486 -11.35933486 -5.35933486 -5.35933486 -7.35933486
## 973 974 975 976 977 978
## -0.35933486 9.64066514 10.64066514 -1.35933486 6.64066514 -6.35933486
## 979 980 981 982 983 984
## -9.35933486 4.64066514 0.64066514 8.64066514 6.64066514 -3.35933486
## 985 986 987 988 989 990
## -2.82291176 10.17708824 -1.82291176 2.17708824 -0.82291176 -2.82291176
## 991 992 993 994 995 996
## -3.82291176 1.17708824 -11.82291176 -7.82291176 -11.82291176 8.17708824
## 997 998 999 1000 1001 1002
## 2.17708824 10.17708824 4.17708824 8.17708824 -3.82291176 -1.82291176
## 1003 1004 1005 1006 1007 1008
## 0.17708824 9.17708824 0.17708824 4.17708824 -7.82291176 -1.82291176
## 1009 1010 1011 1012 1013 1014
## -14.70739847 3.29260153 -4.70739847 4.29260153 13.29260153 2.29260153
## 1015 1016 1017 1018 1019 1020
## 6.29260153 -4.70739847 0.29260153 -0.70739847 -2.70739847 -3.70739847
## 1021 1022 1023 1024 1025 1026
## -6.70739847 6.29260153 0.29260153 8.29260153 -1.70739847 6.29260153
## 1027 1028 1029 1030 1031 1032
## 4.29260153 -0.70739847 -5.70739847 -3.70739847 8.29260153 -13.70739847
## 1033 1034 1035 1036 1037 1038
## -8.01619042 -4.01619042 3.98380958 7.98380958 4.98380958 10.98380958
## 1039 1040 1041 1042 1043 1044
## 9.98380958 11.98380958 7.98380958 -6.01619042 -8.01619042 -6.01619042
## 1045 1046 1047 1048 1049 1050
## 3.98380958 3.98380958 -11.01619042 3.98380958 -13.01619042 -10.01619042
## 1051 1052 1053 1054 1055 1056
## 6.98380958 11.98380958 -8.01619042 -8.01619042 -4.01619042 -5.01619042
## 1057 1058 1059 1060 1061 1062
## -2.98367831 3.01632169 -8.98367831 9.01632169 -2.98367831 -0.98367831
## 1063 1064 1065 1066 1067 1068
## 5.01632169 -1.98367831 -5.98367831 9.01632169 -6.98367831 7.01632169
## 1069 1070 1071 1072 1073 1074
## -6.98367831 7.01632169 9.01632169 -5.98367831 9.01632169 0.01632169
## 1075 1076 1077 1078 1079 1080
## -5.98367831 -3.98367831 -10.98367831 1.01632169 3.01632169 -2.98367831
## 1081 1082 1083 1084 1085 1086
## 3.30226476 -11.69773524 8.30226476 -9.69773524 -5.69773524 6.30226476
## 1087 1088 1089 1090 1091 1092
## -5.69773524 -5.69773524 -1.69773524 -7.69773524 6.30226476 -5.69773524
## 1093 1094 1095 1096 1097 1098
## -3.69773524 -0.69773524 6.30226476 3.30226476 -11.69773524 12.30226476
## 1099 1100 1101 1102 1103 1104
## 8.30226476 0.30226476 -0.69773524 9.30226476 4.30226476 -1.69773524
## 1105 1106 1107 1108 1109 1110
## -10.54206213 -4.54206213 7.45793787 0.45793787 -6.54206213 2.45793787
## 1111 1112 1113 1114 1115 1116
## -2.54206213 6.45793787 11.45793787 1.45793787 4.45793787 -3.54206213
## 1117 1118 1119 1120 1121 1122
## -3.54206213 6.45793787 2.45793787 -1.54206213 -14.54206213 2.45793787
## 1123 1124 1125 1126 1127 1128
## 1.45793787 2.45793787 5.45793787 -3.54206213 5.45793787 -2.54206213
## 1129 1130 1131 1132 1133 1134
## -13.86806991 -5.86806991 -7.86806991 -10.86806991 8.13193009 6.13193009
## 1135 1136 1137 1138 1139 1140
## -11.86806991 1.13193009 12.13193009 -1.86806991 1.13193009 -0.86806991
## 1141 1142 1143 1144 1145 1146
## 8.13193009 2.13193009 -9.86806991 10.13193009 6.13193009 -0.86806991
## 1147 1148 1149 1150 1151 1152
## 2.13193009 -0.86806991 -4.86806991 11.13193009 -5.86806991 8.13193009
## 1153 1154 1155 1156 1157 1158
## -5.54091969 9.45908031 2.45908031 -11.54091969 -7.54091969 -1.54091969
## 1159 1160 1161 1162 1163 1164
## -11.54091969 3.45908031 -15.54091969 -7.54091969 -4.54091969 8.45908031
## 1165 1166 1167 1168 1169 1170
## 0.45908031 4.45908031 7.45908031 6.45908031 -1.54091969 1.45908031
## 1171 1172 1173 1174 1175 1176
## 11.45908031 0.45908031 3.45908031 6.45908031 6.45908031 1.45908031
## 1177 1178 1179 1180 1181 1182
## -13.51640470 4.48359530 -8.51640470 2.48359530 -5.51640470 -9.51640470
## 1183 1184 1185 1186 1187 1188
## -4.51640470 4.48359530 -5.51640470 0.48359530 6.48359530 -11.51640470
## 1189 1190 1191 1192 1193 1194
## 4.48359530 1.48359530 -4.51640470 8.48359530 11.48359530 -12.51640470
## 1195 1196 1197 1198 1199 1200
## 8.48359530 6.48359530 5.48359530 11.48359530 6.48359530 -3.51640470
## 1201 1202 1203 1204 1205 1206
## 6.13016887 6.13016887 9.13016887 6.13016887 -10.86983113 1.13016887
## 1207 1208 1209 1210 1211 1212
## -0.86983113 -1.86983113 6.13016887 -3.86983113 -7.86983113 2.13016887
## 1213 1214 1215 1216 1217 1218
## 4.13016887 -1.86983113 -10.86983113 -8.86983113 -5.86983113 6.13016887
## 1219 1220 1221 1222 1223 1224
## 6.13016887 -7.86983113 12.13016887 2.13016887 -9.86983113 0.13016887
## 1225 1226 1227 1228 1229 1230
## -0.42161372 6.57838628 10.57838628 -2.42161372 -4.42161372 -3.42161372
## 1231 1232 1233 1234 1235 1236
## 10.57838628 -8.42161372 10.57838628 -6.42161372 -2.42161372 10.57838628
## 1237 1238 1239 1240 1241 1242
## 4.57838628 -5.42161372 2.57838628 3.57838628 -5.42161372 -7.42161372
## 1243 1244 1245 1246 1247 1248
## 0.57838628 -4.42161372 -11.42161372 1.57838628 -6.42161372 2.57838628
## 1249 1250 1251 1252 1253 1254
## -5.17800431 -6.17800431 6.82199569 2.82199569 -3.17800431 -5.17800431
## 1255 1256 1257 1258 1259 1260
## -9.17800431 10.82199569 -2.17800431 -5.17800431 8.82199569 2.82199569
## 1261 1262 1263 1264 1265 1266
## 7.82199569 2.82199569 12.82199569 -7.17800431 0.82199569 -1.17800431
## 1267 1268 1269 1270 1271 1272
## -6.17800431 12.82199569 -3.17800431 -7.17800431 -5.17800431 -4.17800431
## 1273 1274 1275 1276 1277 1278
## 9.98952180 -10.01047820 -2.01047820 5.98952180 -3.01047820 -7.01047820
## 1279 1280 1281 1282 1283 1284
## -12.01047820 -7.01047820 0.98952180 -4.01047820 -7.01047820 2.98952180
## 1285 1286 1287 1288 1289 1290
## -7.01047820 6.98952180 0.98952180 -6.01047820 -3.01047820 4.98952180
## 1291 1292 1293 1294 1295 1296
## 0.98952180 1.98952180 9.98952180 13.98952180 11.98952180 -6.01047820
## 1297 1298 1299 1300 1301 1302
## -3.76634511 1.23365489 3.23365489 10.23365489 9.23365489 -7.76634511
## 1303 1304 1305 1306 1307 1308
## -9.76634511 -11.76634511 7.23365489 6.23365489 -2.76634511 -4.76634511
## 1309 1310 1311 1312 1313 1314
## 2.23365489 3.23365489 -7.76634511 -1.76634511 1.23365489 3.23365489
## 1315 1316 1317 1318 1319 1320
## -10.76634511 3.23365489 -2.76634511 -6.76634511 10.23365489 6.23365489
## 1321 1322 1323 1324 1325 1326
## 4.32581158 -9.67418842 -5.67418842 -3.67418842 -7.67418842 -9.67418842
## 1327 1328 1329 1330 1331 1332
## -1.67418842 14.32581158 -1.67418842 -4.67418842 0.32581158 0.32581158
## 1333 1334 1335 1336 1337 1338
## 12.32581158 6.32581158 11.32581158 9.32581158 -1.67418842 9.32581158
## 1339 1340 1341 1342 1343 1344
## 1.32581158 -1.67418842 -3.67418842 -4.67418842 -9.67418842 0.32581158
## 1345 1346 1347 1348 1349 1350
## 5.40760711 -8.59239289 4.40760711 5.40760711 3.40760711 -0.59239289
## 1351 1352 1353 1354 1355 1356
## -4.59239289 -0.59239289 11.40760711 -2.59239289 -4.59239289 -14.59239289
## 1357 1358 1359 1360 1361 1362
## 6.40760711 7.40760711 1.40760711 -4.59239289 -7.59239289 5.40760711
## 1363 1364 1365 1366 1367 1368
## 7.40760711 -7.59239289 6.40760711 2.40760711 -7.59239289 4.40760711
## 1369 1370 1371 1372 1373 1374
## 10.92486294 10.92486294 -5.07513706 -2.07513706 -3.07513706 -5.07513706
## 1375 1376 1377 1378 1379 1380
## -9.07513706 -9.07513706 1.92486294 -9.07513706 -7.07513706 -8.07513706
## 1381 1382 1383 1384 1385 1386
## 0.92486294 9.92486294 -9.07513706 -9.07513706 -5.07513706 0.92486294
## 1387 1388 1389 1390 1391 1392
## -4.07513706 14.92486294 10.92486294 10.92486294 12.92486294 -6.07513706
## 1393 1394 1395 1396 1397 1398
## -0.47581632 -8.47581632 -8.47581632 7.52418368 1.52418368 -1.47581632
## 1399 1400 1401 1402 1403 1404
## -8.47581632 -2.47581632 -10.47581632 7.52418368 1.52418368 0.52418368
## 1405 1406 1407 1408 1409 1410
## 6.52418368 1.52418368 -3.47581632 2.52418368 3.52418368 -1.47581632
## 1411 1412 1413 1414 1415 1416
## 1.52418368 -4.47581632 7.52418368 -0.47581632 13.52418368 -1.47581632
## 1417 1418 1419 1420 1421 1422
## -5.57535134 2.42464866 0.42464866 -2.57535134 6.42464866 -7.57535134
## 1423 1424 1425 1426 1427 1428
## 3.42464866 12.42464866 10.42464866 -7.57535134 8.42464866 7.42464866
## 1429 1430 1431 1432 1433 1434
## 0.42464866 -12.57535134 2.42464866 -12.57535134 0.42464866 14.42464866
## 1435 1436 1437 1438 1439 1440
## -1.57535134 4.42464866 -11.57535134 -1.57535134 -3.57535134 -6.57535134
## 1441 1442 1443 1444 1445 1446
## -2.77310467 -7.77310467 -1.77310467 -10.77310467 -12.77310467 -0.77310467
## 1447 1448 1449 1450 1451 1452
## 2.22689533 3.22689533 7.22689533 1.22689533 -1.77310467 3.22689533
## 1453 1454 1455 1456 1457 1458
## 5.22689533 8.22689533 -1.77310467 -4.77310467 -0.77310467 1.22689533
## 1459 1460 1461 1462 1463 1464
## -10.77310467 -0.77310467 6.22689533 14.22689533 6.22689533 3.22689533
## 1465 1466 1467 1468 1469 1470
## -2.69078547 7.30921453 4.30921453 -6.69078547 2.30921453 -5.69078547
## 1471 1472 1473 1474 1475 1476
## -1.69078547 1.30921453 10.30921453 -4.69078547 -3.69078547 -4.69078547
## 1477 1478 1479 1480 1481 1482
## -2.69078547 11.30921453 0.30921453 4.30921453 7.30921453 -3.69078547
## 1483 1484 1485 1486 1487 1488
## 6.30921453 -3.69078547 -0.69078547 9.30921453 -7.69078547 -11.69078547
## 1489 1490 1491 1492 1493 1494
## -4.48143343 -8.48143343 -11.48143343 -8.48143343 9.51856657 7.51856657
## 1495 1496 1497 1498 1499 1500
## 8.51856657 0.51856657 4.51856657 -9.48143343 -7.48143343 10.51856657
## 1501 1502 1503 1504 1505 1506
## 0.51856657 7.51856657 8.51856657 2.51856657 0.51856657 1.51856657
## 1507 1508 1509 1510 1511 1512
## -12.48143343 6.51856657 8.51856657 -3.48143343 -2.48143343 2.51856657
## 1513 1514 1515 1516 1517 1518
## -9.25337373 3.74662627 -4.25337373 2.74662627 -11.25337373 7.74662627
## 1519 1520 1521 1522 1523 1524
## 4.74662627 -4.25337373 -3.25337373 -10.25337373 -1.25337373 11.74662627
## 1525 1526 1527 1528 1529 1530
## -6.25337373 4.74662627 4.74662627 1.74662627 -6.25337373 1.74662627
## 1531 1532 1533 1534 1535 1536
## 5.74662627 3.74662627 -8.25337373 3.74662627 5.74662627 8.74662627
## 1537 1538 1539 1540 1541 1542
## 3.96395948 -6.03604052 -1.03604052 0.96395948 8.96395948 4.96395948
## 1543 1544 1545 1546 1547 1548
## -1.03604052 0.96395948 -11.03604052 7.96395948 -2.03604052 9.96395948
## 1549 1550 1551 1552 1553 1554
## 7.96395948 9.96395948 -2.03604052 -4.03604052 1.96395948 -6.03604052
## 1555 1556 1557 1558 1559 1560
## -12.03604052 -6.03604052 7.96395948 -2.03604052 -2.03604052 -1.03604052
## 1561 1562 1563 1564 1565 1566
## -4.85093325 4.14906675 4.14906675 -1.85093325 -9.85093325 10.14906675
## 1567 1568 1569 1570 1571 1572
## -9.85093325 5.14906675 4.14906675 -3.85093325 0.14906675 2.14906675
## 1573 1574 1575 1576 1577 1578
## 5.14906675 11.14906675 -5.85093325 7.14906675 10.14906675 -4.85093325
## 1579 1580 1581 1582 1583 1584
## -3.85093325 4.14906675 -11.85093325 -7.85093325 -0.85093325 -0.85093325
## 1585 1586 1587 1588 1589 1590
## -7.54723474 8.45276526 -7.54723474 10.45276526 4.45276526 2.45276526
## 1591 1592 1593 1594 1595 1596
## -3.54723474 -5.54723474 13.45276526 -0.54723474 8.45276526 -10.54723474
## 1597 1598 1599 1600 1601 1602
## -3.54723474 -0.54723474 -6.54723474 -1.54723474 -5.54723474 3.45276526
## 1603 1604 1605 1606 1607 1608
## 10.45276526 2.45276526 -7.54723474 2.45276526 5.45276526 -3.54723474
## 1609 1610 1611 1612 1613 1614
## -5.50147376 -0.50147376 -8.50147376 -4.50147376 -6.50147376 -0.50147376
## 1615 1616 1617 1618 1619 1620
## 8.49852624 -0.50147376 1.49852624 11.49852624 6.49852624 14.49852624
## 1621 1622 1623 1624 1625 1626
## 9.49852624 -4.50147376 -0.50147376 -8.50147376 3.49852624 -3.50147376
## 1627 1628 1629 1630 1631 1632
## 8.49852624 -0.50147376 -1.50147376 -6.50147376 -6.50147376 1.49852624
## 1633 1634 1635 1636 1637 1638
## 3.63522231 8.63522231 -5.36477769 8.63522231 4.63522231 -2.36477769
## 1639 1640 1641 1642 1643 1644
## 8.63522231 6.63522231 4.63522231 -3.36477769 -7.36477769 6.63522231
## 1645 1646 1647 1648 1649 1650
## -2.36477769 5.63522231 -1.36477769 -6.36477769 6.63522231 -10.36477769
## 1651 1652 1653 1654 1655 1656
## -7.36477769 2.63522231 -7.36477769 -8.36477769 -5.36477769 10.63522231
## 1657 1658 1659 1660 1661 1662
## -0.43505367 6.56494633 6.56494633 -6.43505367 3.56494633 3.56494633
## 1663 1664 1665 1666 1667 1668
## 3.56494633 -8.43505367 -11.43505367 -6.43505367 1.56494633 5.56494633
## 1669 1670 1671 1672 1673 1674
## 7.56494633 -4.43505367 -6.43505367 -8.43505367 1.56494633 -4.43505367
## 1675 1676 1677 1678 1679 1680
## 1.56494633 -0.43505367 9.56494633 0.56494633 12.56494633 7.56494633
## 1681 1682 1683 1684 1685 1686
## 8.86670934 -7.13329066 0.86670934 2.86670934 8.86670934 -0.13329066
## 1687 1688 1689 1690 1691 1692
## -5.13329066 -1.13329066 2.86670934 1.86670934 4.86670934 -4.13329066
## 1693 1694 1695 1696 1697 1698
## 0.86670934 -7.13329066 8.86670934 8.86670934 -5.13329066 3.86670934
## 1699 1700 1701 1702 1703 1704
## 6.86670934 -3.13329066 -6.13329066 -7.13329066 -5.13329066 -3.13329066
## 1705 1706 1707 1708 1709 1710
## -1.56946485 0.43053515 3.43053515 3.43053515 -4.56946485 8.43053515
## 1711 1712 1713 1714 1715 1716
## 12.43053515 -11.56946485 -8.56946485 -11.56946485 -6.56946485 -4.56946485
## 1717 1718 1719 1720 1721 1722
## -0.56946485 11.43053515 -0.56946485 -3.56946485 0.43053515 -6.56946485
## 1723 1724 1725 1726 1727 1728
## 9.43053515 -2.56946485 11.43053515 -7.56946485 11.43053515 9.43053515
## 1729 1730 1731 1732 1733 1734
## 8.83183319 0.83183319 5.83183319 9.83183319 5.83183319 1.83183319
## 1735 1736 1737 1738 1739 1740
## -8.16816681 0.83183319 -10.16816681 -3.16816681 4.83183319 -6.16816681
## 1741 1742 1743 1744 1745 1746
## -3.16816681 -8.16816681 -0.16816681 -3.16816681 10.83183319 5.83183319
## 1747 1748 1749 1750 1751 1752
## -7.16816681 4.83183319 -10.16816681 1.83183319 -5.16816681 8.83183319
## 1753 1754 1755 1756 1757 1758
## -4.90523095 12.09476905 -5.90523095 -2.90523095 -0.90523095 -8.90523095
## 1759 1760 1761 1762 1763 1764
## 4.09476905 -6.90523095 -6.90523095 -1.90523095 -9.90523095 2.09476905
## 1765 1766 1767 1768 1769 1770
## 12.09476905 -0.90523095 8.09476905 -6.90523095 2.09476905 13.09476905
## 1771 1772 1773 1774 1775 1776
## -4.90523095 2.09476905 5.09476905 9.09476905 2.09476905 -8.90523095
## 1777 1778 1779 1780 1781 1782
## -3.42996023 -0.42996023 9.57003977 -9.42996023 -10.42996023 8.57003977
## 1783 1784 1785 1786 1787 1788
## 2.57003977 4.57003977 -1.42996023 1.57003977 -13.42996023 13.57003977
## 1789 1790 1791 1792 1793 1794
## 10.57003977 -0.42996023 1.57003977 -2.42996023 -1.42996023 -9.42996023
## 1795 1796 1797 1798 1799 1800
## 1.57003977 -2.42996023 7.57003977 7.57003977 -3.42996023 -0.42996023
## 1801 1802 1803 1804 1805 1806
## 5.25210827 -6.74789173 -3.74789173 9.25210827 12.25210827 4.25210827
## 1807 1808 1809 1810 1811 1812
## 11.25210827 -2.74789173 0.25210827 -0.74789173 5.25210827 -7.74789173
## 1813 1814 1815 1816 1817 1818
## -2.74789173 11.25210827 -6.74789173 3.25210827 3.25210827 0.25210827
## 1819 1820 1821 1822 1823 1824
## -10.74789173 3.25210827 -4.74789173 -11.74789173 6.25210827 -8.74789173
## 1825 1826 1827 1828 1829 1830
## 8.84668496 8.84668496 8.84668496 4.84668496 8.84668496 -1.15331504
## 1831 1832 1833 1834 1835 1836
## 1.84668496 -9.15331504 -12.15331504 3.84668496 -5.15331504 -8.15331504
## 1837 1838 1839 1840 1841 1842
## 8.84668496 4.84668496 -11.15331504 -8.15331504 -3.15331504 12.84668496
## 1843 1844 1845 1846 1847 1848
## 9.84668496 4.84668496 -7.15331504 -11.15331504 -11.15331504 6.84668496
## 1849 1850 1851 1852 1853 1854
## 5.27908242 -11.72091758 -7.72091758 -1.72091758 10.27908242 11.27908242
## 1855 1856 1857 1858 1859 1860
## 12.27908242 10.27908242 4.27908242 0.27908242 -2.72091758 3.27908242
## 1861 1862 1863 1864 1865 1866
## 0.27908242 -14.72091758 4.27908242 -9.72091758 9.27908242 -11.72091758
## 1867 1868 1869 1870 1871 1872
## -1.72091758 -4.72091758 -9.72091758 10.27908242 8.27908242 2.27908242
## 1873 1874 1875 1876 1877 1878
## 10.59226988 0.59226988 -2.40773012 -5.40773012 -3.40773012 -2.40773012
## 1879 1880 1881 1882 1883 1884
## 0.59226988 -11.40773012 1.59226988 7.59226988 -3.40773012 0.59226988
## 1885 1886 1887 1888 1889 1890
## -2.40773012 -3.40773012 -1.40773012 9.59226988 1.59226988 7.59226988
## 1891 1892 1893 1894 1895 1896
## -7.40773012 -0.40773012 5.59226988 5.59226988 -6.40773012 5.59226988
## 1897 1898 1899 1900 1901 1902
## -4.65809909 2.34190091 -8.65809909 2.34190091 11.34190091 4.34190091
## 1903 1904 1905 1906 1907 1908
## 9.34190091 -5.65809909 -13.65809909 -5.65809909 8.34190091 13.34190091
## 1909 1910 1911 1912 1913 1914
## 6.34190091 -10.65809909 6.34190091 -5.65809909 -3.65809909 -11.65809909
## 1915 1916 1917 1918 1919 1920
## 7.34190091 9.34190091 -10.65809909 7.34190091 -6.65809909 11.34190091
## 1921 1922 1923 1924 1925 1926
## -7.73013629 -5.73013629 7.26986371 7.26986371 13.26986371 -1.73013629
## 1927 1928 1929 1930 1931 1932
## 5.26986371 6.26986371 -4.73013629 -1.73013629 -6.73013629 -8.73013629
## 1933 1934 1935 1936 1937 1938
## 6.26986371 -0.73013629 1.26986371 -8.73013629 10.26986371 -4.73013629
## 1939 1940 1941 1942 1943 1944
## 6.26986371 2.26986371 6.26986371 4.26986371 -8.73013629 -3.73013629
## 1945 1946 1947 1948 1949 1950
## 1.57460955 3.57460955 -8.42539045 -6.42539045 9.57460955 -6.42539045
## 1951 1952 1953 1954 1955 1956
## -0.42539045 -1.42539045 0.57460955 5.57460955 10.57460955 1.57460955
## 1957 1958 1959 1960 1961 1962
## 3.57460955 -1.42539045 -6.42539045 3.57460955 5.57460955 -11.42539045
## 1963 1964 1965 1966 1967 1968
## 2.57460955 9.57460955 -4.42539045 -4.42539045 -5.42539045 9.57460955
## 1969 1970 1971 1972 1973 1974
## -2.30757632 5.69242368 -8.30757632 -5.30757632 -5.30757632 7.69242368
## 1975 1976 1977 1978 1979 1980
## 6.69242368 5.69242368 0.69242368 -11.30757632 8.69242368 -3.30757632
## 1981 1982 1983 1984 1985 1986
## 8.69242368 -9.30757632 -8.30757632 1.69242368 10.69242368 12.69242368
## 1987 1988 1989 1990 1991 1992
## -12.30757632 8.69242368 2.69242368 2.69242368 -1.30757632 -1.30757632
## 1993 1994 1995 1996 1997 1998
## -2.21384863 -0.21384863 -9.21384863 10.78615137 5.78615137 -10.21384863
## 1999 2000 2001 2002 2003 2004
## -9.21384863 -7.21384863 -7.21384863 11.78615137 5.78615137 -0.21384863
## 2005 2006 2007 2008 2009 2010
## 7.78615137 -4.21384863 -4.21384863 7.78615137 2.78615137 8.78615137
## 2011 2012 2013 2014 2015 2016
## -2.21384863 3.78615137 5.78615137 -4.21384863 9.78615137 -10.21384863
## 2017 2018 2019 2020 2021 2022
## -5.74779662 5.25220338 -2.74779662 13.25220338 -2.74779662 1.25220338
## 2023 2024 2025 2026 2027 2028
## -1.74779662 -7.74779662 -6.74779662 8.25220338 -8.74779662 -3.74779662
## 2029 2030 2031 2032 2033 2034
## -4.74779662 -3.74779662 1.25220338 -10.74779662 1.25220338 7.25220338
## 2035 2036 2037 2038 2039 2040
## 13.25220338 3.25220338 11.25220338 9.25220338 -10.74779662 11.25220338
## 2041 2042 2043 2044 2045 2046
## 0.75417888 -10.24582112 9.75417888 9.75417888 9.75417888 -6.24582112
## 2047 2048 2049 2050 2051 2052
## 5.75417888 2.75417888 -6.24582112 -4.24582112 -12.24582112 5.75417888
## 2053 2054 2055 2056 2057 2058
## 9.75417888 3.75417888 -0.24582112 3.75417888 4.75417888 3.75417888
## 2059 2060 2061 2062 2063 2064
## -10.24582112 -8.24582112 -3.24582112 5.75417888 -2.24582112 1.75417888
## 2065 2066 2067 2068 2069 2070
## 11.36885912 -8.63114088 13.36885912 3.36885912 -11.63114088 7.36885912
## 2071 2072 2073 2074 2075 2076
## 5.36885912 5.36885912 -8.63114088 13.36885912 -6.63114088 -6.63114088
## 2077 2078 2079 2080 2081 2082
## 3.36885912 -4.63114088 -11.63114088 3.36885912 -1.63114088 1.36885912
## 2083 2084 2085 2086 2087 2088
## 7.36885912 10.36885912 1.36885912 3.36885912 -8.63114088 -6.63114088
## 2089 2090 2091 2092 2093 2094
## 1.11444403 -9.88555597 10.11444403 2.11444403 -1.88555597 -0.88555597
## 2095 2096 2097 2098 2099 2100
## 6.11444403 -6.88555597 12.11444403 0.11444403 -0.88555597 -4.88555597
## 2101 2102 2103 2104 2105 2106
## 13.11444403 -5.88555597 3.11444403 -6.88555597 -5.88555597 2.11444403
## 2107 2108 2109 2110 2111 2112
## 4.11444403 14.11444403 10.11444403 -7.88555597 -7.88555597 -2.88555597
## 2113 2114 2115 2116 2117 2118
## -3.52721036 -1.52721036 -5.52721036 5.47278964 -0.52721036 2.47278964
## 2119 2120 2121 2122 2123 2124
## 15.47278964 4.47278964 3.47278964 7.47278964 -1.52721036 -3.52721036
## 2125 2126 2127 2128 2129 2130
## 10.47278964 -9.52721036 0.47278964 7.47278964 0.47278964 -4.52721036
## 2131 2132 2133 2134 2135 2136
## -7.52721036 -6.52721036 -4.52721036 -7.52721036 10.47278964 -4.52721036
## 2137 2138 2139 2140 2141 2142
## 0.61000937 -3.38999063 -9.38999063 -3.38999063 0.61000937 5.61000937
## 2143 2144 2145 2146 2147 2148
## -5.38999063 -5.38999063 -5.38999063 6.61000937 -2.38999063 7.61000937
## 2149 2150 2151 2152 2153 2154
## -6.38999063 4.61000937 -6.38999063 -1.38999063 7.61000937 10.61000937
## 2155 2156 2157 2158 2159 2160
## -5.38999063 2.61000937 -3.38999063 14.61000937 -7.38999063 10.61000937
## 2161 2162 2163 2164 2165 2166
## -1.78627438 2.21372562 0.21372562 14.21372562 -1.78627438 -1.78627438
## 2167 2168 2169 2170 2171 2172
## 0.21372562 6.21372562 -0.78627438 3.21372562 -5.78627438 7.21372562
## 2173 2174 2175 2176 2177 2178
## -5.78627438 -5.78627438 -7.78627438 2.21372562 -5.78627438 -8.78627438
## 2179 2180 2181 2182 2183 2184
## -0.78627438 15.21372562 -1.78627438 -5.78627438 12.21372562 -3.78627438
## 2185 2186 2187 2188 2189 2190
## -9.96909677 0.03090323 -6.96909677 -2.96909677 9.03090323 -5.96909677
## 2191 2192 2193 2194 2195 2196
## 8.03090323 15.03090323 -5.96909677 1.03090323 0.03090323 7.03090323
## 2197 2198 2199 2200 2201 2202
## 12.03090323 1.03090323 7.03090323 -8.96909677 -12.96909677 3.03090323
## 2203 2204 2205 2206 2207 2208
## -1.96909677 4.03090323 -2.96909677 -2.96909677 7.03090323 0.03090323
## 2209 2210 2211 2212 2213 2214
## -0.61286178 2.38713822 -6.61286178 1.38713822 -7.61286178 -8.61286178
## 2215 2216 2217 2218 2219 2220
## -9.61286178 3.38713822 -9.61286178 -0.61286178 -0.61286178 15.38713822
## 2221 2222 2223 2224 2225 2226
## 3.38713822 -6.61286178 9.38713822 -1.61286178 13.38713822 3.38713822
## 2227 2228 2229 2230 2231 2232
## 2.38713822 11.38713822 -13.61286178 7.38713822 -5.61286178 13.38713822
## 2233 2234 2235 2236 2237 2238
## 4.83543479 7.83543479 8.83543479 2.83543479 4.83543479 -10.16456521
## 2239 2240 2241 2242 2243 2244
## -0.16456521 1.83543479 11.83543479 1.83543479 3.83543479 -0.16456521
## 2245 2246 2247 2248 2249 2250
## -8.16456521 1.83543479 -2.16456521 -6.16456521 8.83543479 2.83543479
## 2251 2252 2253 2254 2255 2256
## -8.16456521 13.83543479 -12.16456521 -1.16456521 1.83543479 -12.16456521
## 2257 2258 2259 2260 2261 2262
## 2.49186180 -1.50813820 -6.50813820 -7.50813820 2.49186180 -2.50813820
## 2263 2264 2265 2266 2267 2268
## 5.49186180 -3.50813820 10.49186180 6.49186180 7.49186180 -5.50813820
## 2269 2270 2271 2272 2273 2274
## 8.49186180 -0.50813820 3.49186180 4.49186180 -1.50813820 -11.50813820
## 2275 2276 2277 2278 2279 2280
## 2.49186180 1.49186180 -3.50813820 9.49186180 4.49186180 -2.50813820
## 2281 2282 2283 2284 2285 2286
## -9.40474729 0.59525271 8.59525271 6.59525271 -7.40474729 -3.40474729
## 2287 2288 2289 2290 2291 2292
## -1.40474729 -0.40474729 -3.40474729 -4.40474729 -9.40474729 11.59525271
## 2293 2294 2295 2296 2297 2298
## 14.59525271 3.59525271 0.59525271 8.59525271 -0.40474729 6.59525271
## 2299 2300 2301 2302 2303 2304
## -4.40474729 -1.40474729 -5.40474729 -0.40474729 4.59525271 -0.40474729
## 2305 2306 2307 2308 2309 2310
## 5.79608397 10.79608397 -12.20391603 -5.20391603 -11.20391603 0.79608397
## 2311 2312 2313 2314 2315 2316
## -5.20391603 4.79608397 -13.20391603 -4.20391603 8.79608397 4.79608397
## 2317 2318 2319 2320 2321 2322
## 5.79608397 5.79608397 12.79608397 -3.20391603 10.79608397 0.79608397
## 2323 2324 2325 2326 2327 2328
## 3.79608397 -5.20391603 -3.20391603 -1.20391603 8.79608397 4.79608397
## 2329 2330 2331 2332 2333 2334
## 7.81602919 6.81602919 3.81602919 0.81602919 -0.18397081 11.81602919
## 2335 2336 2337 2338 2339 2340
## -8.18397081 7.81602919 -0.18397081 -8.18397081 4.81602919 -8.18397081
## 2341 2342 2343 2344 2345 2346
## 7.81602919 1.81602919 -8.18397081 4.81602919 -14.18397081 1.81602919
## 2347 2348 2349 2350 2351 2352
## -8.18397081 7.81602919 8.81602919 -4.18397081 7.81602919 -3.18397081
## 2353 2354 2355 2356 2357 2358
## -0.13249761 -8.13249761 12.86750239 -8.13249761 -9.13249761 7.86750239
## 2359 2360 2361 2362 2363 2364
## -1.13249761 1.86750239 5.86750239 6.86750239 -11.13249761 -0.13249761
## 2365 2366 2367 2368 2369 2370
## 5.86750239 -2.13249761 1.86750239 9.86750239 5.86750239 7.86750239
## 2371 2372 2373 2374 2375 2376
## -6.13249761 -0.13249761 9.86750239 -2.13249761 -6.13249761 -1.13249761
## 2377 2378 2379 2380 2381 2382
## -5.20453481 2.79546519 -7.20453481 -9.20453481 -1.20453481 0.79546519
## 2383 2384 2385 2386 2387 2388
## -1.20453481 0.79546519 6.79546519 -11.20453481 7.79546519 7.79546519
## 2389 2390 2391 2392 2393 2394
## 8.79546519 0.79546519 4.79546519 4.79546519 -8.20453481 8.79546519
## 2395 2396 2397 2398 2399 2400
## 5.79546519 8.79546519 -6.20453481 -10.20453481 8.79546519 2.79546519
## 2401 2402 2403 2404 2405 2406
## -0.90039179 -8.90039179 -2.90039179 -2.90039179 13.09960821 -7.90039179
## 2407 2408 2409 2410 2411 2412
## 11.09960821 -2.90039179 -8.90039179 -11.90039179 2.09960821 -2.90039179
## 2413 2414 2415 2416 2417 2418
## 5.09960821 1.09960821 4.09960821 -6.90039179 3.09960821 14.09960821
## 2419 2420 2421 2422 2423 2424
## -0.90039179 7.09960821 4.09960821 11.09960821 3.09960821 -0.90039179
## 2425 2426 2427 2428 2429 2430
## 11.28525344 -5.71474656 -0.71474656 13.28525344 -0.71474656 3.28525344
## 2431
## 6.28525344
VarCorr(condmodel1) #shows random variances, square rooted as sds
## Groups Name Std.Dev.
## BowlingTeamNo (Intercept) 3.5816
## Residual 6.9482
vcov(condmodel1) #shows covariance of fixed effects (not sure this is interesting)
## 2 x 2 Matrix of class "dpoMatrix"
## (Intercept) GroupGripStrength
## (Intercept) 3.79707065 -0.073056883
## GroupGripStrength -0.07305688 0.001461783
df.residual(condmodel1) #shows df of residual
## [1] 2427
head(class06)
## ï..const ID BowlingTeamNo GroupGripStrength PercentWinBowl
## 1 1 1 1 65 51
## 2 1 2 1 65 51
## 3 1 3 1 65 46
## 4 1 4 1 65 50
## 5 1 5 1 65 62
## 6 1 6 1 65 45
## ForearmLengthQuartile WinningOrientation pred_0 pred_1
## 1 2 52 43.83257 43.81571
## 2 2 43 43.83257 43.81571
## 3 1 42 43.83257 43.81571
## 4 1 44 43.83257 43.81571
## 5 4 35 43.83257 43.81571
## 6 1 49 43.83257 43.81571
### 2. Conditional model: Fixed effect of Level 1 PercentWinBowl
condmodel2 = lmer(WinningOrientation ~ GroupGripStrength + PercentWinBowl +
(1|BowlingTeamNo), data=class06, REML=FALSE)
summary(condmodel2)
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: WinningOrientation ~ GroupGripStrength + PercentWinBowl + (1 |
## BowlingTeamNo)
## Data: class06
##
## AIC BIC logLik deviance df.resid
## 16278.8 16307.8 -8134.4 16268.8 2426
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.11822 -0.85455 -0.00689 0.80870 2.17507
##
## Random effects:
## Groups Name Variance Std.Dev.
## BowlingTeamNo (Intercept) 12.55 3.543
## Residual 43.28 6.579
## Number of obs: 2431, groups: BowlingTeamNo, 102
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 36.39438 2.04084 17.833
## GroupGripStrength -0.11419 0.03814 -2.993
## PercentWinBowl 0.33729 0.02044 16.502
##
## Correlation of Fixed Effects:
## (Intr) GrpGrS
## GrpGrpStrng -0.850
## PercntWnBwl -0.344 -0.168
#lower case anova does nested model tests
anova(nullmodel,condmodel1,condmodel2)
## Data: class06
## Models:
## nullmodel: WinningOrientation ~ 1 + (1 | BowlingTeamNo)
## condmodel1: WinningOrientation ~ GroupGripStrength + (1 | BowlingTeamNo)
## condmodel2: WinningOrientation ~ GroupGripStrength + PercentWinBowl + (1 |
## condmodel2: BowlingTeamNo)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## nullmodel 3 16533 16550 -8263.4 16527
## condmodel1 4 16535 16558 -8263.3 16527 0.0486 1 0.8256
## condmodel2 5 16279 16308 -8134.4 16269 257.8588 1 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#Saving predicted values to the data frame, so we can plot what the conditional model looks like
pred_2<-(fitted(condmodel2))
pred2<-as.data.frame(pred_2)
class06<-cbind(class06,pred2)
#This plot will show the overall fixed intercept and slope;
qplot(y=pred_2, x=PercentWinBowl, colour = BowlingTeamNo,
data = class06)
#The second plot imposes the best fitting line
ggplot(class06) +
aes(x=class06$PercentWinBowl, y=class06$pred_2) + #, col=as.factor(hsb_pred2$school)
geom_point() +
stat_smooth(method = "lm", se = FALSE)
## Warning: Use of `class06$PercentWinBowl` is discouraged. Use `PercentWinBowl`
## instead.
## Warning: Use of `class06$pred_2` is discouraged. Use `pred_2` instead.
## Warning: Use of `class06$PercentWinBowl` is discouraged. Use `PercentWinBowl`
## instead.
## Warning: Use of `class06$pred_2` is discouraged. Use `pred_2` instead.
## `geom_smooth()` using formula 'y ~ x'
#Gives you each school's fixed and random! coefficients
coef(condmodel2)
## $BowlingTeamNo
## (Intercept) GroupGripStrength PercentWinBowl
## 1 33.30907 -0.1141861 0.3372947
## 2 32.21007 -0.1141861 0.3372947
## 3 32.00927 -0.1141861 0.3372947
## 4 30.34880 -0.1141861 0.3372947
## 5 31.65271 -0.1141861 0.3372947
## 6 32.24810 -0.1141861 0.3372947
## 7 29.68188 -0.1141861 0.3372947
## 8 31.00859 -0.1141861 0.3372947
## 9 31.88875 -0.1141861 0.3372947
## 10 32.52095 -0.1141861 0.3372947
## 11 33.24026 -0.1141861 0.3372947
## 12 34.91523 -0.1141861 0.3372947
## 13 30.98308 -0.1141861 0.3372947
## 14 30.68238 -0.1141861 0.3372947
## 15 33.48213 -0.1141861 0.3372947
## 16 33.61841 -0.1141861 0.3372947
## 17 34.38295 -0.1141861 0.3372947
## 18 33.29434 -0.1141861 0.3372947
## 19 33.68652 -0.1141861 0.3372947
## 20 33.80854 -0.1141861 0.3372947
## 21 30.57338 -0.1141861 0.3372947
## 22 35.50234 -0.1141861 0.3372947
## 23 33.65743 -0.1141861 0.3372947
## 24 30.92926 -0.1141861 0.3372947
## 25 32.43196 -0.1141861 0.3372947
## 26 33.34927 -0.1141861 0.3372947
## 27 34.88512 -0.1141861 0.3372947
## 28 31.68446 -0.1141861 0.3372947
## 29 32.62862 -0.1141861 0.3372947
## 30 35.60601 -0.1141861 0.3372947
## 31 36.52333 -0.1141861 0.3372947
## 32 35.40551 -0.1141861 0.3372947
## 33 33.19628 -0.1141861 0.3372947
## 34 33.96647 -0.1141861 0.3372947
## 35 31.71059 -0.1141861 0.3372947
## 36 35.73746 -0.1141861 0.3372947
## 37 34.80651 -0.1141861 0.3372947
## 38 32.54602 -0.1141861 0.3372947
## 39 33.36368 -0.1141861 0.3372947
## 40 34.51212 -0.1141861 0.3372947
## 41 34.45371 -0.1141861 0.3372947
## 42 36.43140 -0.1141861 0.3372947
## 43 36.75114 -0.1141861 0.3372947
## 44 35.79780 -0.1141861 0.3372947
## 45 35.54722 -0.1141861 0.3372947
## 46 35.41309 -0.1141861 0.3372947
## 47 38.14528 -0.1141861 0.3372947
## 48 37.26659 -0.1141861 0.3372947
## 49 38.24512 -0.1141861 0.3372947
## 50 37.71346 -0.1141861 0.3372947
## 51 35.49539 -0.1141861 0.3372947
## 52 35.15341 -0.1141861 0.3372947
## 53 36.26256 -0.1141861 0.3372947
## 54 35.94064 -0.1141861 0.3372947
## 55 35.87149 -0.1141861 0.3372947
## 56 36.84543 -0.1141861 0.3372947
## 57 37.60830 -0.1141861 0.3372947
## 58 35.27451 -0.1141861 0.3372947
## 59 37.46197 -0.1141861 0.3372947
## 60 36.23168 -0.1141861 0.3372947
## 61 37.32335 -0.1141861 0.3372947
## 62 38.21049 -0.1141861 0.3372947
## 63 39.59123 -0.1141861 0.3372947
## 64 38.44317 -0.1141861 0.3372947
## 65 38.92718 -0.1141861 0.3372947
## 66 37.31417 -0.1141861 0.3372947
## 67 37.60872 -0.1141861 0.3372947
## 68 37.52515 -0.1141861 0.3372947
## 69 38.92944 -0.1141861 0.3372947
## 70 39.66755 -0.1141861 0.3372947
## 71 38.32666 -0.1141861 0.3372947
## 72 38.64726 -0.1141861 0.3372947
## 73 38.22946 -0.1141861 0.3372947
## 74 36.94512 -0.1141861 0.3372947
## 75 39.12210 -0.1141861 0.3372947
## 76 38.05818 -0.1141861 0.3372947
## 77 38.39686 -0.1141861 0.3372947
## 78 40.85912 -0.1141861 0.3372947
## 79 38.39076 -0.1141861 0.3372947
## 80 40.13435 -0.1141861 0.3372947
## 81 39.87543 -0.1141861 0.3372947
## 82 39.21426 -0.1141861 0.3372947
## 83 40.05129 -0.1141861 0.3372947
## 84 38.91580 -0.1141861 0.3372947
## 85 40.18282 -0.1141861 0.3372947
## 86 40.57904 -0.1141861 0.3372947
## 87 40.00027 -0.1141861 0.3372947
## 88 39.69925 -0.1141861 0.3372947
## 89 38.71819 -0.1141861 0.3372947
## 90 37.69079 -0.1141861 0.3372947
## 91 37.84513 -0.1141861 0.3372947
## 92 40.19851 -0.1141861 0.3372947
## 93 40.68838 -0.1141861 0.3372947
## 94 41.36698 -0.1141861 0.3372947
## 95 41.98640 -0.1141861 0.3372947
## 96 40.29931 -0.1141861 0.3372947
## 97 43.25310 -0.1141861 0.3372947
## 98 43.16997 -0.1141861 0.3372947
## 99 41.39829 -0.1141861 0.3372947
## 100 41.64319 -0.1141861 0.3372947
## 101 41.69271 -0.1141861 0.3372947
## 102 43.20921 -0.1141861 0.3372947
##
## attr(,"class")
## [1] "coef.mer"
library(car)
#Gives you chi-square for each model term
Anova(condmodel2)
## Analysis of Deviance Table (Type II Wald chisquare tests)
##
## Response: WinningOrientation
## Chisq Df Pr(>Chisq)
## GroupGripStrength 8.961 1 0.002758 **
## PercentWinBowl 272.330 1 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#This ensures that your model is stored in a lmerMod object, needed for the next step
class(condmodel2) <- "lmerMod"
#Stargazer makes a nicely formated model summary table (albeit incomplete)
stargazer(condmodel2, type = "text",
digits = 3,
star.cutoffs = c(0.05, 0.01, 0.001),
digit.separator = "")
##
## =================================================
## Dependent variable:
## -----------------------------
## WinningOrientation
## -------------------------------------------------
## GroupGripStrength -0.114**
## (0.038)
##
## PercentWinBowl 0.337***
## (0.020)
##
## Constant 36.394***
## (2.041)
##
## -------------------------------------------------
## Observations 2431
## Log Likelihood -8134.405
## Akaike Inf. Crit. 16278.810
## Bayesian Inf. Crit. 16307.790
## =================================================
## Note: *p<0.05; **p<0.01; ***p<0.001
###sjPlot::tab_model is even better, because it formats the table in APA style and adds
# fixed and random effects and p-values
tab_model(nullmodel,condmodel1,condmodel2)
|
|
WinningOrientation
|
WinningOrientation
|
WinningOrientation
|
|
Predictors
|
Estimates
|
CI
|
p
|
Estimates
|
CI
|
p
|
Estimates
|
CI
|
p
|
|
(Intercept)
|
47.55
|
46.80 – 48.30
|
<0.001
|
47.97
|
44.15 – 51.79
|
<0.001
|
36.39
|
32.39 – 40.39
|
<0.001
|
|
GroupGripStrength
|
|
|
|
-0.01
|
-0.08 – 0.07
|
0.826
|
-0.11
|
-0.19 – -0.04
|
0.003
|
|
PercentWinBowl
|
|
|
|
|
|
|
0.34
|
0.30 – 0.38
|
<0.001
|
|
Random Effects
|
|
σ2
|
48.28
|
48.28
|
43.28
|
|
Ï„00
|
12.84 BowlingTeamNo
|
12.83 BowlingTeamNo
|
12.55 BowlingTeamNo
|
|
ICC
|
0.21
|
0.21
|
0.22
|
|
N
|
102 BowlingTeamNo
|
102 BowlingTeamNo
|
102 BowlingTeamNo
|
|
Observations
|
2431
|
2431
|
2431
|
|
Marginal R2 / Conditional R2
|
0.000 / 0.210
|
0.000 / 0.210
|
0.083 / 0.289
|
#Exact p-values and df in revised syntax below, but warning VERY time consuming (crashed my computer)
#tab_model(condmodel2, p.val = "kr", show.df = TRUE)
### 3. Conditional model: Random effect of Level 1 PercentWinBowl
#Double line creates a "variance components" (no correlated random effects) solution like SPSS
condmodel3 = lmer(WinningOrientation ~ GroupGripStrength + PercentWinBowl +
(1 + PercentWinBowl||BowlingTeamNo), data=class06, REML=FALSE)
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.205288 (tol = 0.002, component 1)
summary(condmodel3)
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: WinningOrientation ~ GroupGripStrength + PercentWinBowl + ((1 |
## BowlingTeamNo) + (0 + PercentWinBowl | BowlingTeamNo))
## Data: class06
##
## AIC BIC logLik deviance df.resid
## 16280.6 16315.3 -8134.3 16268.6 2425
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.11992 -0.85186 -0.00686 0.81319 2.17117
##
## Random effects:
## Groups Name Variance Std.Dev.
## BowlingTeamNo (Intercept) 1.140e+01 3.37573
## BowlingTeamNo.1 PercentWinBowl 5.179e-04 0.02276
## Residual 4.324e+01 6.57549
## Number of obs: 2431, groups: BowlingTeamNo, 102
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 36.30849 2.04284 17.774
## GroupGripStrength -0.11255 0.03832 -2.937
## PercentWinBowl 0.33739 0.02056 16.411
##
## Correlation of Fixed Effects:
## (Intr) GrpGrS
## GrpGrpStrng -0.852
## PercntWnBwl -0.340 -0.168
## convergence code: 0
## Model failed to converge with max|grad| = 0.205288 (tol = 0.002, component 1)
#lower case anova does nested model tests
anova(nullmodel,condmodel1,condmodel2,condmodel3)
## Data: class06
## Models:
## nullmodel: WinningOrientation ~ 1 + (1 | BowlingTeamNo)
## condmodel1: WinningOrientation ~ GroupGripStrength + (1 | BowlingTeamNo)
## condmodel2: WinningOrientation ~ GroupGripStrength + PercentWinBowl + (1 |
## condmodel2: BowlingTeamNo)
## condmodel3: WinningOrientation ~ GroupGripStrength + PercentWinBowl + ((1 |
## condmodel3: BowlingTeamNo) + (0 + PercentWinBowl | BowlingTeamNo))
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## nullmodel 3 16533 16550 -8263.4 16527
## condmodel1 4 16535 16558 -8263.3 16527 0.0486 1 0.8256
## condmodel2 5 16279 16308 -8134.4 16269 257.8588 1 <2e-16 ***
## condmodel3 6 16281 16315 -8134.3 16269 0.2370 1 0.6264
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#Saving predicted values to the data frame, so we can plot what the conditional model looks like
pred_3<-(fitted(condmodel3))
pred3<-as.data.frame(pred_3)
class06<-cbind(class06,pred3)
#This plot will show the overall fixed intercept and slope;
qplot(y=pred_3, x=PercentWinBowl, colour = BowlingTeamNo,
data = class06)
#The second plot imposes the best fitting line
ggplot(class06) +
aes(x=class06$PercentWinBowl, y=class06$pred_3) + #, col=as.factor(hsb_pred2$school)
geom_point() +
stat_smooth(method = "lm", se = FALSE)
## Warning: Use of `class06$PercentWinBowl` is discouraged. Use `PercentWinBowl`
## instead.
## Warning: Use of `class06$pred_3` is discouraged. Use `pred_3` instead.
## Warning: Use of `class06$PercentWinBowl` is discouraged. Use `PercentWinBowl`
## instead.
## Warning: Use of `class06$pred_3` is discouraged. Use `pred_3` instead.
## `geom_smooth()` using formula 'y ~ x'
#Gives you each school's fixed and random! coefficients
coef(condmodel3)
## $BowlingTeamNo
## (Intercept) GroupGripStrength PercentWinBowl
## 1 33.68778 -0.1125499 0.3275568
## 2 32.45829 -0.1125499 0.3308502
## 3 32.38390 -0.1125499 0.3283138
## 4 31.01419 -0.1125499 0.3229822
## 5 31.94857 -0.1125499 0.3299413
## 6 32.63025 -0.1125499 0.3275950
## 7 30.25606 -0.1125499 0.3238819
## 8 31.43791 -0.1125499 0.3271424
## 9 32.39586 -0.1125499 0.3253085
## 10 32.64605 -0.1125499 0.3333769
## 11 33.38061 -0.1125499 0.3329848
## 12 34.93591 -0.1125499 0.3350786
## 13 31.53952 -0.1125499 0.3248026
## 14 31.25249 -0.1125499 0.3239621
## 15 33.66187 -0.1125499 0.3319977
## 16 33.78451 -0.1125499 0.3323513
## 17 34.60124 -0.1125499 0.3306051
## 18 33.53272 -0.1125499 0.3308676
## 19 33.95347 -0.1125499 0.3298851
## 20 33.81380 -0.1125499 0.3362752
## 21 31.35713 -0.1125499 0.3208101
## 22 35.54434 -0.1125499 0.3346468
## 23 33.87791 -0.1125499 0.3308381
## 24 31.26467 -0.1125499 0.3289678
## 25 32.87050 -0.1125499 0.3257683
## 26 33.54567 -0.1125499 0.3314900
## 27 34.99595 -0.1125499 0.3334187
## 28 31.84297 -0.1125499 0.3330473
## 29 32.80961 -0.1125499 0.3321650
## 30 35.59356 -0.1125499 0.3354940
## 31 36.45956 -0.1125499 0.3373582
## 32 35.40110 -0.1125499 0.3355918
## 33 33.27875 -0.1125499 0.3343504
## 34 34.35650 -0.1125499 0.3273556
## 35 31.90644 -0.1125499 0.3320476
## 36 35.74842 -0.1125499 0.3352453
## 37 34.75424 -0.1125499 0.3369756
## 38 32.91125 -0.1125499 0.3280480
## 39 33.63637 -0.1125499 0.3297219
## 40 34.54268 -0.1125499 0.3350054
## 41 34.44298 -0.1125499 0.3365081
## 42 36.39785 -0.1125499 0.3360069
## 43 36.58117 -0.1125499 0.3402080
## 44 35.75193 -0.1125499 0.3369900
## 45 35.68114 -0.1125499 0.3322929
## 46 35.59798 -0.1125499 0.3312932
## 47 38.06283 -0.1125499 0.3373179
## 48 37.09020 -0.1125499 0.3393444
## 49 38.04771 -0.1125499 0.3401815
## 50 37.38672 -0.1125499 0.3427717
## 51 35.52092 -0.1125499 0.3352722
## 52 35.23340 -0.1125499 0.3338776
## 53 36.17353 -0.1125499 0.3373552
## 54 35.88408 -0.1125499 0.3370431
## 55 35.90732 -0.1125499 0.3348914
## 56 36.70562 -0.1125499 0.3389928
## 57 37.34134 -0.1125499 0.3408442
## 58 35.25812 -0.1125499 0.3362688
## 59 37.39514 -0.1125499 0.3365381
## 60 36.11819 -0.1125499 0.3379427
## 61 37.25858 -0.1125499 0.3367763
## 62 38.03323 -0.1125499 0.3388662
## 63 39.23069 -0.1125499 0.3420431
## 64 38.15589 -0.1125499 0.3406348
## 65 38.60054 -0.1125499 0.3427188
## 66 37.04282 -0.1125499 0.3407610
## 67 37.30629 -0.1125499 0.3411771
## 68 37.41226 -0.1125499 0.3380884
## 69 38.77557 -0.1125499 0.3386540
## 70 39.27105 -0.1125499 0.3435019
## 71 38.10336 -0.1125499 0.3403384
## 72 38.09233 -0.1125499 0.3471556
## 73 37.90741 -0.1125499 0.3426678
## 74 36.78472 -0.1125499 0.3390559
## 75 38.73058 -0.1125499 0.3439780
## 76 37.77176 -0.1125499 0.3418158
## 77 37.89458 -0.1125499 0.3458429
## 78 40.21023 -0.1125499 0.3493307
## 79 38.22952 -0.1125499 0.3382782
## 80 39.55839 -0.1125499 0.3464901
## 81 39.52430 -0.1125499 0.3415989
## 82 38.87854 -0.1125499 0.3423873
## 83 39.57045 -0.1125499 0.3442026
## 84 38.49250 -0.1125499 0.3447974
## 85 39.55080 -0.1125499 0.3477731
## 86 40.13245 -0.1125499 0.3442535
## 87 39.45158 -0.1125499 0.3484043
## 88 39.46385 -0.1125499 0.3406438
## 89 38.46537 -0.1125499 0.3408213
## 90 37.54961 -0.1125499 0.3386087
## 91 37.48486 -0.1125499 0.3423702
## 92 39.73536 -0.1125499 0.3455699
## 93 40.04880 -0.1125499 0.3491116
## 94 40.88657 -0.1125499 0.3462033
## 95 41.28184 -0.1125499 0.3484560
## 96 39.74820 -0.1125499 0.3477616
## 97 42.25971 -0.1125499 0.3562596
## 98 42.40134 -0.1125499 0.3507223
## 99 40.97936 -0.1125499 0.3438286
## 100 41.15355 -0.1125499 0.3454873
## 101 41.03635 -0.1125499 0.3481243
## 102 42.40961 -0.1125499 0.3517951
##
## attr(,"class")
## [1] "coef.mer"
#Gives you chi-square for each model term
Anova(condmodel3)
## Analysis of Deviance Table (Type II Wald chisquare tests)
##
## Response: WinningOrientation
## Chisq Df Pr(>Chisq)
## GroupGripStrength 8.6244 1 0.003317 **
## PercentWinBowl 269.3114 1 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#This ensures that your model is stored in a lmerMod object, needed for the next step
class(condmodel3) <- "lmerMod"
#Stargazer makes a nicely formated model summary table (albeit incomplete)
stargazer(condmodel3, type = "text",
digits = 3,
star.cutoffs = c(0.05, 0.01, 0.001),
digit.separator = "")
##
## =================================================
## Dependent variable:
## -----------------------------
## WinningOrientation
## -------------------------------------------------
## GroupGripStrength -0.113**
## (0.038)
##
## PercentWinBowl 0.337***
## (0.021)
##
## Constant 36.308***
## (2.043)
##
## -------------------------------------------------
## Observations 2431
## Log Likelihood -8134.286
## Akaike Inf. Crit. 16280.570
## Bayesian Inf. Crit. 16315.350
## =================================================
## Note: *p<0.05; **p<0.01; ***p<0.001
###sjPlot::tab_model is even better, because it formats the table in APA style and adds
# fixed and random effects and p-values
tab_model(nullmodel,condmodel1,condmodel2,condmodel3)
|
|
WinningOrientation
|
WinningOrientation
|
WinningOrientation
|
WinningOrientation
|
|
Predictors
|
Estimates
|
CI
|
p
|
Estimates
|
CI
|
p
|
Estimates
|
CI
|
p
|
Estimates
|
CI
|
p
|
|
(Intercept)
|
47.55
|
46.80 – 48.30
|
<0.001
|
47.97
|
44.15 – 51.79
|
<0.001
|
36.39
|
32.39 – 40.39
|
<0.001
|
36.31
|
32.30 – 40.31
|
<0.001
|
|
GroupGripStrength
|
|
|
|
-0.01
|
-0.08 – 0.07
|
0.826
|
-0.11
|
-0.19 – -0.04
|
0.003
|
-0.11
|
-0.19 – -0.04
|
0.003
|
|
PercentWinBowl
|
|
|
|
|
|
|
0.34
|
0.30 – 0.38
|
<0.001
|
0.34
|
0.30 – 0.38
|
<0.001
|
|
Random Effects
|
|
σ2
|
48.28
|
48.28
|
43.28
|
43.24
|
|
Ï„00
|
12.84 BowlingTeamNo
|
12.83 BowlingTeamNo
|
12.55 BowlingTeamNo
|
11.40 BowlingTeamNo
|
|
|
|
|
0.00 BowlingTeamNo.1
|
|
ICC
|
0.21
|
0.21
|
0.22
|
0.21
|
|
N
|
102 BowlingTeamNo
|
102 BowlingTeamNo
|
102 BowlingTeamNo
|
102 BowlingTeamNo
|
|
Observations
|
2431
|
2431
|
2431
|
2431
|
|
Marginal R2 / Conditional R2
|
0.000 / 0.210
|
0.000 / 0.210
|
0.083 / 0.289
|
0.085 / 0.276
|
#Exact p-values and df in revised syntax below, but warning VERY time consuming (crashed my computer)
#tab_model(condmodel2, p.val = "kr", show.df = TRUE)
library(lattice)
# This is an alternative to the qplot above; shows each school in a truly unique color
with(class06, xyplot(pred_3 ~ PercentWinBowl, group=BowlingTeamNo))
### 4. Repeating the above model series in lme
###### Trying lme
# nlme::lme is an older function, generally superseded by lme4:lmer
# It offers less control over random covariance, etc
# It does include df, t, p-values though
library(nlme)
##
## Attaching package: 'nlme'
## The following object is masked from 'package:lme4':
##
## lmList
nullmodel2 <- lme(WinningOrientation~1,
data=class06,
random=~1|BowlingTeamNo,
na.action=na.omit,
method="ML",
control=list(maxIter=100,msMaxIter=100,tolerance=1e-5,
opt="optim",optimMethod="BFGS"))
summary(nullmodel2)
## Linear mixed-effects model fit by maximum likelihood
## Data: class06
## AIC BIC logLik
## 16532.72 16550.1 -8263.358
##
## Random effects:
## Formula: ~1 | BowlingTeamNo
## (Intercept) Residual
## StdDev: 3.582684 6.948192
##
## Fixed effects: WinningOrientation ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 47.54993 0.3820665 2329 124.4546 0
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.23424567 -0.78093979 -0.02999008 0.78812321 2.49793568
##
## Number of Observations: 2431
## Number of Groups: 102
condmodel1b <- lme(WinningOrientation~GroupGripStrength,
data=class06,
random=~1|BowlingTeamNo,
na.action=na.omit,
method="ML",
control=list(maxIter=100,msMaxIter=100,tolerance=1e-5,
opt="optim",optimMethod="BFGS"))
summary(condmodel1b)
## Linear mixed-effects model fit by maximum likelihood
## Data: class06
## AIC BIC logLik
## 16534.67 16557.85 -8263.334
##
## Random effects:
## Formula: ~1 | BowlingTeamNo
## (Intercept) Residual
## StdDev: 3.581624 6.948197
##
## Fixed effects: WinningOrientation ~ GroupGripStrength
## Value Std.Error DF t-value p-value
## (Intercept) 47.97113 1.949410 2329 24.608031 0.0000
## GroupGripStrength -0.00843 0.038249 100 -0.220345 0.8261
## Correlation:
## (Intr)
## GroupGripStrength -0.981
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.23668369 -0.77876057 -0.02976581 0.78739686 2.49475687
##
## Number of Observations: 2431
## Number of Groups: 102
condmodel2b <- lme(WinningOrientation~GroupGripStrength + PercentWinBowl,
data=class06,
random=~1|BowlingTeamNo,
na.action=na.omit,
method="ML",
control=list(maxIter=100,msMaxIter=100,tolerance=1e-5,
opt="optim",optimMethod="BFGS"))
summary(condmodel2b)
## Linear mixed-effects model fit by maximum likelihood
## Data: class06
## AIC BIC logLik
## 16278.81 16307.79 -8134.405
##
## Random effects:
## Formula: ~1 | BowlingTeamNo
## (Intercept) Residual
## StdDev: 3.542689 6.578827
##
## Fixed effects: WinningOrientation ~ GroupGripStrength + PercentWinBowl
## Value Std.Error DF t-value p-value
## (Intercept) 36.39438 2.0420959 2328 17.822071 0.0000
## GroupGripStrength -0.11419 0.0381684 100 -2.991637 0.0035
## PercentWinBowl 0.33729 0.0204517 2328 16.492253 0.0000
## Correlation:
## (Intr) GrpGrS
## GroupGripStrength -0.850
## PercentWinBowl -0.344 -0.168
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.118222355 -0.854554256 -0.006894971 0.808704013 2.175067336
##
## Number of Observations: 2431
## Number of Groups: 102
condmodel3b <- lme(WinningOrientation~GroupGripStrength + PercentWinBowl,
data=class06,
random=~1 + PercentWinBowl|BowlingTeamNo,
na.action=na.omit,
method="ML",
control=list(maxIter=100,msMaxIter=100,tolerance=1e-5,
opt="optim",optimMethod="BFGS"))
summary(condmodel3b)
## Linear mixed-effects model fit by maximum likelihood
## Data: class06
## AIC BIC logLik
## 16282.62 16323.19 -8134.311
##
## Random effects:
## Formula: ~1 + PercentWinBowl | BowlingTeamNo
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 3.73448604 (Intr)
## PercentWinBowl 0.05155614 -0.401
## Residual 6.56726968
##
## Fixed effects: WinningOrientation ~ GroupGripStrength + PercentWinBowl
## Value Std.Error DF t-value p-value
## (Intercept) 36.21924 2.0552976 2328 17.622383 0.0000
## GroupGripStrength -0.11088 0.0385389 100 -2.877128 0.0049
## PercentWinBowl 0.33757 0.0210720 2328 16.019600 0.0000
## Correlation:
## (Intr) GrpGrS
## GroupGripStrength -0.851
## PercentWinBowl -0.345 -0.165
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.123416658 -0.849702508 -0.007557133 0.811058706 2.159672462
##
## Number of Observations: 2431
## Number of Groups: 102
anova(nullmodel2,condmodel1b,condmodel2b,condmodel3b)
## Model df AIC BIC logLik Test L.Ratio p-value
## nullmodel2 1 3 16532.72 16550.10 -8263.358
## condmodel1b 2 4 16534.67 16557.85 -8263.334 1 vs 2 0.04858 0.8256
## condmodel2b 3 5 16278.81 16307.79 -8134.405 2 vs 3 257.85879 <.0001
## condmodel3b 4 7 16282.62 16323.19 -8134.311 3 vs 4 0.18699 0.9107