I'm pretty new in using
lmer and be confused about different p-values in Tukey post hoc tests associated with exactly the same estimates. I built a linear mixed model with monetary contributions of human subjects as response variable and their wealth and number of children as explanatory variables. The experiment was designed in a way to contribute for future generations. I don't have repeated measurements of the same individual but some individuals played within the same group. There are several subsets and additional random factors but here I only want to consider the following model where
totalcontSubject means contribution of a subject over the entire game,
poverty is a factor with 2 levels (rich and poor), and
children is a factor with 2 levels (child or noChild). Particularly I'm interested in understanding the fixed effects part of the model.
> summary(TC1) Linear mixed model fit by REML ['lmerMod'] Formula: totalcontSubject ~ poverty * children + (1 | group_2) Data: data REML criterion at convergence: 414.6 Scaled residuals: Min 1Q Median 3Q Max -2.42955 -0.45554 -0.09361 0.45228 2.33159 Random effects: Groups Name Variance Std.Dev. group_2 (Intercept) 8.611e-15 9.280e-08 Residual 1.042e+02 1.021e+01 Number of obs: 58, groups: group_2, 10 Fixed effects: Estimate Std. Error t value (Intercept) 16.200 3.228 5.019 povertyrich 8.600 3.953 2.175 childrennoChild 2.800 4.565 0.613 povertyrich:childrennoChild -4.489 5.642 -0.796 Correlation of Fixed Effects: (Intr) pvrty chldrC povertyrch -0.816 chldrnnChld -0.707 0.577 pvrtyrch:C 0.572 -0.701 -0.809
If I interpret fixed effects of the summary table in the right way, my intercept denotes poor people with children. The estimate also corresponds to the mean value of this combination in my data. According to my calculations the difference to rich people (shown as
povertyrich) actually shows the difference of the intercept to rich people with children, even if not explicitly mentioned by
povertyrich. This is the first issue I'm a bit confused. A reduced model only with fixed factor poverty is significant better by
anova() but it seems data including children are used for this evaluation.
If I run a Tukey post hoc test by means of my TC1 model, I get a significant difference between rich and poor. But the estimates in the summary actually include children. Estimates of intercept and slope are the means of poor people with children and the difference to rich people with children. They don't correspond to the means of poor or rich data irrespective of parenthood.
summary(glht(TC1, linfct=mcp(povertry="Tukey"))) Simultaneous Tests for General Linear Hypotheses Multiple Comparisons of Means: Tukey Contrasts Fit: lmer(formula = totalcontSubject ~ poverty * children + (1 | group_2), data = data) Linear Hypotheses: Estimate Std. Error z value Pr(>|z|) rich - poor == 0 8.600 3.953 2.175 0.0296 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Adjusted p values reported -- single-step method)
I get even more confused when I run a Tukey post hoc test for a subset where I coded interactions in a column such as poor people with children and rich people with children. In this output I have exactly the same estimates and parameters for these categories (like in the summary shown before for rich poor people exclusively) but the p-values are different. A visual check indicates that there is a significant difference between
poorChild but outputs of
Interak shows me it is not. Also, a comparison between models
anova() with fixed factor poverty vs. fixed factors poverty and children indicates that I can get rid of the variable children in my model. Before I do so, I would like to understand the outputs better. I also worry about the high value for Residual and the correlations in the summary table.
> summary(glht(TC1_2, linfct=mcp(Interak="Tukey"))) Simultaneous Tests for General Linear Hypotheses Multiple Comparisons of Means: Tukey Contrasts Fit: lmer(formula = totalcontSubject ~ Interak + (1 | group_2), data = data) Linear Hypotheses: Estimate Std. Error z value Pr(>|z|) poorNoChild - poorChild == 0 2.800 4.565 0.613 0.927 richChild - poorChild == 0 8.600 3.953 2.175 0.129 richNoChild - poorChild == 0 6.911 4.026 1.717 0.312 richChild - poorNoChild == 0 5.800 3.953 1.467 0.454 richNoChild - poorNoChild == 0 4.111 4.026 1.021 0.735 richNoChild - richChild == 0 -1.689 3.316 -0.509 0.956 (Adjusted p values reported -- single-step method)