## 'data.frame': 12 obs. of 2 variables:
## $ family : int 1 1 1 2 2 2 2 3 3 4 ...
## $ liberalism: int 25 29 34 18 21 21 26 31 35 30 ...## family liberalism
## 1 1 25
## 2 1 29
## 3 1 34
## 4 2 18
## 5 2 21
## 6 2 21## Df Sum Sq Mean Sq F value Pr(>F)
## family 3 244.33 81.44 7.726 0.00951 **
## Residuals 8 84.33 10.54
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1##
## Error: family
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 3 244.3 81.44
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 8 84.33 10.54##
## Call:
## lm(formula = liberalism ~ family - 1, data = dta)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.333 -1.000 -0.500 1.500 4.667
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## family1 29.333 1.875 15.65 2.77e-07 ***
## family2 21.500 1.623 13.24 1.01e-06 ***
## family3 33.000 2.296 14.37 5.36e-07 ***
## family4 30.667 1.875 16.36 1.96e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.247 on 8 degrees of freedom
## Multiple R-squared: 0.9911, Adjusted R-squared: 0.9867
## F-statistic: 223.6 on 4 and 8 DF, p-value: 3.065e-08## Analysis of Variance Table
##
## Response: liberalism
## Df Sum Sq Mean Sq F value Pr(>F)
## family 4 9429.7 2357.42 223.63 3.065e-08 ***
## Residuals 8 84.3 10.54
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## Linear mixed model fit by REML ['lmerMod']
## Formula: liberalism ~ 1 + (1 | family)
## Data: dta
##
## REML criterion at convergence: 65.4
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.3116 -0.3850 -0.1135 0.5998 1.4772
##
## Random effects:
## Groups Name Variance Std.Dev.
## family (Intercept) 21.92 4.682
## Residual 10.48 3.238
## Number of obs: 12, groups: family, 4
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 28.49 2.53 11.26## $family
## (Intercept)
## 1 0.7271474
## 2 -6.2437983
## 3 3.6394896
## 4 1.8771613
##
## with conditional variances for "family"#
# augument data with group mean and ID
dta_fy <- dta %>%
mutate( ID = paste0("S", 11:22), g_ave = mean(liberalism )) %>%
group_by(family) %>%
mutate( grp_ave = mean(liberalism) ) %>%
as.data.frame %>%
mutate( grp_ave_uw = mean(unique(grp_ave)) )
# output from model m3
dta_m3 <- data.frame(ID = paste0("S",11:22), fe = rep(summary(m3)$coef[1], 12),
re = rep(as.numeric(t(as.data.frame(ranef(m3)[1]))), c(3, 4, 2, 3)),
pred = fitted(m3))
head(dta_m3)## ID fe re pred
## 1 S11 28.49027 0.7271474 29.21742
## 2 S12 28.49027 0.7271474 29.21742
## 3 S13 28.49027 0.7271474 29.21742
## 4 S14 28.49027 -6.2437983 22.24648
## 5 S15 28.49027 -6.2437983 22.24648
## 6 S16 28.49027 -6.2437983 22.24648## ID family liberalism g_ave grp_ave grp_ave_uw fe re
## 1 S11 1 25 27.66667 29.33333 28.625 28.49027 0.7271474
## 2 S12 1 29 27.66667 29.33333 28.625 28.49027 0.7271474
## 3 S13 1 34 27.66667 29.33333 28.625 28.49027 0.7271474
## 4 S14 2 18 27.66667 21.50000 28.625 28.49027 -6.2437983
## 5 S15 2 21 27.66667 21.50000 28.625 28.49027 -6.2437983
## 6 S16 2 21 27.66667 21.50000 28.625 28.49027 -6.2437983
## 7 S17 2 26 27.66667 21.50000 28.625 28.49027 -6.2437983
## 8 S18 3 31 27.66667 33.00000 28.625 28.49027 3.6394896
## 9 S19 3 35 27.66667 33.00000 28.625 28.49027 3.6394896
## 10 S20 4 30 27.66667 30.66667 28.625 28.49027 1.8771613
## 11 S21 4 30 27.66667 30.66667 28.625 28.49027 1.8771613
## 12 S22 4 32 27.66667 30.66667 28.625 28.49027 1.8771613
## pred
## 1 29.21742
## 2 29.21742
## 3 29.21742
## 4 22.24648
## 5 22.24648
## 6 22.24648
## 7 22.24648
## 8 32.12976
## 9 32.12976
## 10 30.36744
## 11 30.36744
## 12 30.36744The correlation between any two liberalism ratings from the same family is 21.919/(21.919+10.482) = 0.677
The best linear unbiased prediction of the random effect for family 4 is (30.667 - 28.49)*(21.919/(21.919+(10.482/3))) = 1.877
The predicted mean for family 4 is 28.49 + 1.877 = 30.368 < 30.667