> tapply(score,bowler,mean); tapply(score,bowler,sd) 1 2 3 4 5 6 7 8 209.6607 198.8750 212.9464 216.1786 208.1964 198.6250 209.5357 208.4286 9 10 11 12 13 14 15 198.9643 209.3036 202.4286 194.7500 202.7321 211.5893 205.7143 1 2 3 4 5 6 7 8 31.95922 25.70749 23.45861 26.07340 24.17995 27.14511 29.69418 25.86529 9 10 11 12 13 14 15 23.72798 28.92619 20.68364 26.53043 25.46691 23.57331 21.85382 > tapply(score,pattern,mean); tapply(score,pattern,sd) 1 2 3 4 210.0905 204.5714 201.3952 207.3905 1 2 3 4 28.81315 24.45711 24.32910 26.68566 > tapply(score,list(bowler,pattern),mean); tapply(score,list(bowler,pattern),sd) 1 2 3 4 1 223.2857 208.7857 195.5714 211.0000 2 211.7857 195.7143 196.4286 191.5714 3 218.1429 208.5000 209.6429 215.5000 4 219.4286 216.6429 212.4286 216.2143 5 210.5714 198.4286 204.7143 219.0714 6 211.0000 203.2857 193.0714 187.1429 7 223.3571 199.2857 194.4286 221.0714 8 209.5714 214.2143 208.6429 201.2857 9 199.5714 198.5714 193.2857 204.4286 10 205.8571 213.7143 198.3571 219.2857 11 202.5000 205.2857 194.3571 207.5714 12 206.2143 182.6429 196.1429 194.0000 13 198.5000 207.8571 210.7143 193.8571 14 212.0000 205.8571 208.2857 220.2143 15 199.5714 209.7857 204.8571 208.6429 1 2 3 4 1 42.43625 22.98220 27.26710 28.84974 2 28.77279 27.05672 24.30801 19.77455 3 24.50701 26.23489 27.77826 14.09173 4 18.71533 21.74187 27.51982 35.62449 5 20.77245 18.80145 28.69583 24.81769 6 30.90930 16.89983 23.72426 30.82421 7 34.25831 30.28691 20.24357 22.84215 8 25.16626 31.63537 20.76094 25.93907 9 27.98273 21.66782 18.56974 26.78034 10 33.01715 20.73061 31.22525 27.83862 11 26.88079 14.48834 21.39952 17.66663 12 31.98360 25.79984 16.73254 26.59381 13 15.50558 22.38769 25.87969 33.63802 14 30.02819 25.48561 22.43085 13.00993 15 27.98273 23.93317 20.26690 13.61903 > > wpba.mod1 <- aov(score ~ bowler + pattern + bowler:pattern) > summary(wpba.mod1) Df Sum Sq Mean Sq F value Pr(>F) bowler 14 29965 2140.3 3.295 3.84e-05 *** pattern 3 8785 2928.4 4.508 0.00382 ** bowler:pattern 42 34423 819.6 1.262 0.12711 Residuals 780 506679 649.6 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > wpba.mod2 <- aov(score ~ bowler + pattern + Error(bowler:pattern)) Warning message: In aov(score ~ bowler + pattern + Error(bowler:pattern)) : Error() model is singular > summary(wpba.mod2) Error: bowler:pattern Df Sum Sq Mean Sq F value Pr(>F) bowler 14 29965 2140.3 2.611 0.00824 ** pattern 3 8785 2928.4 3.573 0.02172 * Residuals 42 34423 819.6 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Error: Within Df Sum Sq Mean Sq F value Pr(>F) Residuals 780 506679 649.6 > > library(nlme) > wpba.mod3 <- lme(fixed = score ~ pattern, random = ~1|bowler/pattern) > summary(wpba.mod3) Linear mixed-effects model fit by REML Data: NULL AIC BIC logLik 7848.53 7881.631 -3917.265 Random effects: Formula: ~1 | bowler (Intercept) StdDev: 4.856505 Formula: ~1 | pattern %in% bowler (Intercept) Residual StdDev: 3.48493 25.48701 Fixed effects: score ~ pattern Value Std.Error DF t-value p-value (Intercept) 210.09048 2.339936 780 89.78473 0.0000 pattern2 -5.51905 2.793894 42 -1.97540 0.0548 pattern3 -8.69524 2.793894 42 -3.11223 0.0033 pattern4 -2.70000 2.793894 42 -0.96639 0.3394 Correlation: (Intr) pttrn2 pttrn3 pattern2 -0.597 pattern3 -0.597 0.500 pattern4 -0.597 0.500 0.500 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -2.69109541 -0.66938595 -0.01910369 0.62334718 3.26889501 Number of Observations: 840 Number of Groups: bowler pattern %in% bowler 15 60 > intervals(wpba.mod3) Approximate 95% confidence intervals Fixed effects: lower est. upper (Intercept) 205.497159 210.090476 214.6837932 pattern2 -11.157355 -5.519048 0.1192593 pattern3 -14.333545 -8.695238 -3.0569312 pattern4 -8.338307 -2.700000 2.9383069 attr(,"label") [1] "Fixed effects:" Random Effects: Level: bowler lower est. upper sd((Intercept)) 2.625939 4.856505 8.981791 Level: pattern lower est. upper sd((Intercept)) 1.219097 3.48493 9.962075 Within-group standard error: lower est. upper 24.25308 25.48701 26.78371 > anova(wpba.mod3) numDF denDF F-value p-value (Intercept) 1 780 16631.613 <.0001 pattern 3 42 3.573 0.0217 > > library(lmerTest) Loading required package: Matrix Loading required package: lme4 Attaching package: ‘lme4’ The following object is masked from ‘package:nlme’: lmList Attaching package: ‘lmerTest’ The following object is masked from ‘package:lme4’: lmer The following object is masked from ‘package:stats’: step > wpba.mod4 <- lmer(score~pattern+(1|bowler)+(1|pattern:bowler)) > summary(wpba.mod4) Linear mixed model fit by REML t-tests use Satterthwaite approximations to degrees of freedom [lmerMod] Formula: score ~ pattern + (1 | bowler) + (1 | pattern:bowler) REML criterion at convergence: 7834.5 Scaled residuals: Min 1Q Median 3Q Max -2.6911 -0.6694 -0.0191 0.6234 3.2689 Random effects: Groups Name Variance Std.Dev. pattern:bowler (Intercept) 12.14 3.485 bowler (Intercept) 23.58 4.856 Residual 649.59 25.487 Number of obs: 840, groups: pattern:bowler, 60; bowler, 15 Fixed effects: Estimate Std. Error df t value Pr(>|t|) (Intercept) 210.090 2.340 44.890 89.786 < 2e-16 *** pattern2 -5.519 2.794 42.000 -1.975 0.05482 . pattern3 -8.695 2.794 42.000 -3.112 0.00333 ** pattern4 -2.700 2.794 42.000 -0.966 0.33938 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation of Fixed Effects: (Intr) pttrn2 pttrn3 pattern2 -0.597 pattern3 -0.597 0.500 pattern4 -0.597 0.500 0.500 > anova(wpba.mod4) Analysis of Variance Table of type III with Satterthwaite approximation for degrees of freedom Sum Sq Mean Sq NumDF DenDF F.value Pr(>F) pattern 6962.8 2320.9 3 42 3.5729 0.02172 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > lsmeans(wpba.mod4) Least Squares Means table: pattern Estimate Standard Error DF t-value Lower CI Upper CI pattern 1 1.0 210.09 2.34 44.9 89.79 205 215 pattern 2 2.0 204.57 2.34 44.9 87.43 200 209 pattern 3 3.0 201.40 2.34 44.9 86.07 197 206 pattern 4 4.0 207.39 2.34 44.9 88.63 203 212 p-value pattern 1 <2e-16 *** pattern 2 <2e-16 *** pattern 3 <2e-16 *** pattern 4 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Warning message: 'lsmeans' is deprecated. Use 'lsmeansLT' instead. See help("Deprecated") and help("lmerTest-deprecated"). > difflsmeans(wpba.mod4) Differences of LSMEANS: Estimate Standard Error DF t-value Lower CI Upper CI p-value pattern 1 - 2 5.5 2.79 42.0 1.98 -0.119 11.157 0.055 . pattern 1 - 3 8.7 2.79 42.0 3.11 3.057 14.334 0.003 ** pattern 1 - 4 2.7 2.79 42.0 0.97 -2.938 8.338 0.339 pattern 2 - 3 3.2 2.79 42.0 1.14 -2.462 8.814 0.262 pattern 2 - 4 -2.8 2.79 42.0 -1.01 -8.457 2.819 0.319 pattern 3 - 4 -6.0 2.79 42.0 -2.15 -11.633 -0.357 0.038 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > confint(wpba.mod4) Computing profile confidence intervals ... 2.5 % 97.5 % .sig01 0.000000 6.2461367 .sig02 1.852906 8.1800535 .sigma 24.272735 26.8060450 (Intercept) 205.567564 214.6133882 pattern2 -10.924183 -0.1139124 pattern3 -14.100373 -3.2901029 pattern4 -8.105135 2.7051352 > >