在帮助其他人进行分析时，我遇到了一个关于 lme4 for R 中线性混合模型的 t 检验和 F 检验之间差异的问题，由 lmerTest 提供。我知道为线性混合模型计算任何类型的 p 值的问题（据我了解，主要是因为自由度的定义是有问题的），以及解释主要影响的问题显着交互的存在（基于边际原则）。
简而言之，数据来自具有两个条件（一致性真/假）的实验，在六组传感器上测量，这些传感器可以描述为两个因素的组合：前向性（前/后）和侧向性（左/中/右） .
从下面的总结输出中可以看出，t.tests 没有显示出显着的一致性效应（p = 0.12），而 anova 输出显示出非常显着的一致性效应（p = 2.8e-10）。由于一致性只有两个水平，这不可能是 F 检验对固定因子的多个水平进行综合检验的结果。因此，我不确定是什么导致了方差分析输出中非常显着的结果。这是因为存在涉及一致性的强相互作用，这当然取决于在模型参数化中包含主效应吗？
我已经在 CrossValidated 上寻找了这个问题的先前答案，但除了可能是这个问题的第一个答案之外，我找不到任何相关的东西。但是，如果这确实提供了一个真正的答案，那么它就隐含在数学中，我正在寻找一个概念性的答案，我可以向我试图帮助的人解释。
> final.mod<-lmer(uV~1+factor(congruity)*factor(laterality)*factor(anteriority)+(1|sent.id)+(1|Subject),data=selected.data)
> summary(final.mod)
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: uV ~ 1 + factor(congruity) * factor(laterality) * factor(anteriority) + (1 | sent.id) + (1 | Subject)
Data: selected.data
REML criterion at convergence: 348903.5
Scaled residuals:
Min 1Q Median 3Q Max
-7.0440 -0.6002 0.0069 0.6038 11.3912
Random effects:
Groups Name Variance Std.Dev.
sent.id (Intercept) 1.773 1.332
Subject (Intercept) 2.548 1.596
Residual 111.396 10.554
Number of obs: 46176, groups: sent.id, 41; Subject, 30
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 4.768e-03 3.973e-01 7.900e+01 0.012 0.9905
factor(congruity)TRUE 3.758e-01 2.410e-01 4.611e+04 1.559 0.1189
factor(laterality)left 7.154e-02 2.430e-01 4.610e+04 0.294 0.7685
factor(laterality)right -2.003e-01 2.430e-01 4.610e+04 -0.824 0.4098
factor(anteriority)posterior -4.203e-02 2.430e-01 4.610e+04 -0.173 0.8627
factor(congruity)TRUE:factor(laterality)left -1.013e-01 3.404e-01 4.610e+04 -0.298 0.7660
factor(congruity)TRUE:factor(laterality)right 7.233e-02 3.404e-01 4.610e+04 0.213 0.8317
factor(congruity)TRUE:factor(anteriority)posterior 6.162e-01 3.404e-01 4.610e+04 1.810 0.0702 .
factor(laterality)left:factor(anteriority)posterior 2.568e-01 3.437e-01 4.610e+04 0.747 0.4549
factor(laterality)right:factor(anteriority)posterior 1.763e-01 3.437e-01 4.610e+04 0.513 0.6080
factor(congruity)TRUE:factor(laterality)left:factor(anteriority)posterior -5.162e-02 4.813e-01 4.610e+04 -0.107 0.9146
factor(congruity)TRUE:factor(laterality)right:factor(anteriority)posterior -2.420e-01 4.813e-01 4.610e+04 -0.503 0.6152
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr) fc()TRUE fctr(ltrlty)l fctr(ltrlty)r fctr(n) fctr(cngrty)TRUE:fctr(ltrlty)l fctr(cngrty)TRUE:fctr(ltrlty)r
fctr(c)TRUE -0.310
fctr(ltrlty)l -0.306 0.504
fctr(ltrlty)r -0.306 0.504 0.500
fctr(ntrrt) -0.306 0.504 0.500 0.500
fctr(cngrty)TRUE:fctr(ltrlty)l 0.218 -0.706 -0.714 -0.357 -0.357
fctr(cngrty)TRUE:fctr(ltrlty)r 0.218 -0.706 -0.357 -0.714 -0.357 0.500
fctr(cngrty)TRUE:fctr(n) 0.218 -0.706 -0.357 -0.357 -0.714 0.500 0.500
fctr(ltrlty)l:() 0.216 -0.357 -0.707 -0.354 -0.707 0.505 0.252
fctr(ltrlty)r:() 0.216 -0.357 -0.354 -0.707 -0.707 0.252 0.505
fctr(cngrty)TRUE:fctr(ltrlty)l:() -0.154 0.499 0.505 0.252 0.505 -0.707 -0.354
fctr(cngrty)TRUE:fctr(ltrlty)r:() -0.154 0.499 0.252 0.505 0.505 -0.354 -0.707
fctr(cngrty)TRUE:fctr(n) fctr(ltrlty)l:() fctr(ltrlty)r:() fctr(cngrty)TRUE:fctr(ltrlty)l:()
fctr(c)TRUE
fctr(ltrlty)l
fctr(ltrlty)r
fctr(ntrrt)
fctr(cngrty)TRUE:fctr(ltrlty)l
fctr(cngrty)TRUE:fctr(ltrlty)r
fctr(cngrty)TRUE:fctr(n)
fctr(ltrlty)l:() 0.505
fctr(ltrlty)r:() 0.505 0.500
fctr(cngrty)TRUE:fctr(ltrlty)l:() -0.707 -0.714 -0.357
fctr(cngrty)TRUE:fctr(ltrlty)r:() -0.707 -0.357 -0.714 0.500
> anova(final.mod)
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
factor(congruity) 4439.1 4439.1 1 46142 39.850 2.768e-10 ***
factor(laterality) 572.9 286.5 2 46095 2.572 0.076430 .
factor(anteriority) 1508.1 1508.1 1 46095 13.538 0.000234 ***
factor(congruity):factor(laterality) 31.6 15.8 2 46095 0.142 0.867581
factor(congruity):factor(anteriority) 775.1 775.1 1 46095 6.958 0.008349 **
factor(laterality):factor(anteriority) 111.9 56.0 2 46095 0.502 0.605126
factor(congruity):factor(laterality):factor(anteriority) 31.2 15.6 2 46095 0.140 0.869183
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
回复@Aurelie 的问题：
> congruity.mod<-lmer(uV~1+factor(congruity)+(1|sent.id)+(1|Subject),data=selected.data)
> summary(congruity.mod)
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: uV ~ 1 + factor(congruity) + (1 | sent.id) + (1 | Subject)
Data: selected.data
REML criterion at convergence: 494077.2
Scaled residuals:
Min 1Q Median 3Q Max
-10.1673 -0.5790 -0.0097 0.5818 12.6088
Random effects:
Groups Name Variance Std.Dev.
sent.id (Intercept) 4.568 2.137
Subject (Intercept) 6.132 2.476
Residual 178.137 13.347
Number of obs: 61568, groups: sent.id, 41; Subject, 30
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 0.6055 0.5671 57.0000 1.068 0.29
factor(congruity)FALSE -0.7105 0.1084 61535.0000 -6.558 5.51e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr)
fctr()FALSE -0.093
> anova(congruity.mod)
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
factor(congruity) 7660.5 7660.5 1 61535 43.004 5.507e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> laterality.mod<-lmer(uV~1+factor(laterality)+(1|sent.id)+(1|Subject),data=selected.data)
> summary(laterality.mod)
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: uV ~ 1 + factor(laterality) + (1 | sent.id) + (1 | Subject)
Data: selected.data
REML criterion at convergence: 372848.2
Scaled residuals:
Min 1Q Median 3Q Max
-9.7033 -0.5981 -0.0076 0.6006 12.2265
Random effects:
Groups Name Variance Std.Dev.
sent.id (Intercept) 5.568 2.360
Subject (Intercept) 6.777 2.603
Residual 186.966 13.674
Number of obs: 46176, groups: sent.id, 41; Subject, 30
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 0.8128 0.6115 61.0000 1.329 0.18877
factor(laterality)left -0.4260 0.1559 46105.0000 -2.733 0.00628 **
factor(laterality)right -0.6709 0.1559 46105.0000 -4.304 1.68e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr) fctr(ltrlty)l
fctr(ltrlty)l -0.127
fctr(ltrlty)r -0.127 0.500
> anova(laterality.mod)
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
factor(laterality) 3548.2 1774.1 2 46105 9.4889 7.584e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> anteriority.mod<-lmer(uV~1+factor(anteriority)+(1|sent.id)+(1|Subject),data=selected.data)
> summary(anteriority.mod)
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: uV ~ 1 + factor(anteriority) + (1 | sent.id) + (1 | Subject)
Data: selected.data
REML criterion at convergence: 372738.6
Scaled residuals:
Min 1Q Median 3Q Max
-9.6668 -0.5986 -0.0032 0.6017 12.2711
Random effects:
Groups Name Variance Std.Dev.
sent.id (Intercept) 5.569 2.360
Subject (Intercept) 6.777 2.603
Residual 186.525 13.657
Number of obs: 46176, groups: sent.id, 41; Subject, 30
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -0.2693 0.6081 59.0000 -0.443 0.66
factor(anteriority)posterior 1.4328 0.1271 46105.0000 11.272 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr)
fctr(ntrrt) -0.105
> anova(anteriority.mod)
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
factor(anteriority) 23700 23700 1 46106 127.06 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
更新：根据@Henrik 的回答更新对比后：
> options(contrasts=c("contr.sum","contr.poly"))
> final.mod<-lmer(uV~1+factor(congruity)*factor(laterality)*factor(anteriority)+(1|sent.id)+(1|Subject),data=selected.data)
> summary(final.mod)
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: uV ~ 1 + factor(congruity) * factor(laterality) * factor(anteriority) + (1 | sent.id) + (1 | Subject)
Data: selected.data
REML criterion at convergence: 372689.8
Scaled residuals:
Min 1Q Median 3Q Max
-9.6772 -0.5979 -0.0016 0.5977 12.3439
Random effects:
Groups Name Variance Std.Dev.
sent.id (Intercept) 5.556 2.357
Subject (Intercept) 6.752 2.599
Residual 186.232 13.647
Number of obs: 46176, groups: sent.id, 41; Subject, 30
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 4.355e-01 6.039e-01 5.800e+01 0.721 0.4737
factor(congruity)1 4.501e-01 6.396e-02 4.613e+04 7.037 1.99e-12 ***
factor(laterality)1 3.628e-01 8.983e-02 4.610e+04 4.039 5.38e-05 ***
factor(laterality)2 -5.732e-02 8.983e-02 4.610e+04 -0.638 0.5234
factor(anteriority)1 -7.183e-01 6.352e-02 4.610e+04 -11.308 < 2e-16 ***
factor(congruity)1:factor(laterality)1 1.433e-01 8.983e-02 4.610e+04 1.596 0.1106
factor(congruity)1:factor(laterality)2 -1.535e-01 8.983e-02 4.610e+04 -1.709 0.0875 .
factor(congruity)1:factor(anteriority)1 9.442e-02 6.352e-02 4.610e+04 1.487 0.1371
factor(laterality)1:factor(anteriority)1 2.282e-01 8.983e-02 4.610e+04 2.540 0.0111 *
factor(laterality)2:factor(anteriority)1 -2.121e-01 8.983e-02 4.610e+04 -2.362 0.0182 *
factor(congruity)1:factor(laterality)1:factor(anteriority)1 -7.802e-03 8.983e-02 4.610e+04 -0.087 0.9308
factor(congruity)1:factor(laterality)2:factor(anteriority)1 -1.141e-02 8.983e-02 4.610e+04 -0.127 0.8989
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr) fctr(c)1 fctr(l)1 fct()2 fctr(n)1 fctr(cngrty)1:fctr(l)1 fc()1:()2 fctr(cngrty)1:fctr(n)1
fctr(cngr)1 -0.003
fctr(ltrl)1 0.000 0.000
fctr(ltrl)2 0.000 0.000 -0.500
fctr(ntrr)1 0.000 0.000 0.000 0.000
fctr(cngrty)1:fctr(l)1 0.000 0.000 -0.020 0.010 0.000
fctr()1:()2 0.000 0.000 0.010 -0.020 0.000 -0.500
fctr(cngrty)1:fctr(n)1 0.000 0.000 0.000 0.000 -0.020 0.000 0.000
fctr(l)1:()1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
fctr()2:()1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
f()1:()1:() 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
f()1:()2:() 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
fctr(l)1:()1 f()2:( f()1:()1:
fctr(cngr)1
fctr(ltrl)1
fctr(ltrl)2
fctr(ntrr)1
fctr(cngrty)1:fctr(l)1
fctr()1:()2
fctr(cngrty)1:fctr(n)1
fctr(l)1:()1
fctr()2:()1 -0.500
f()1:()1:() -0.020 0.010
f()1:()2:() 0.010 -0.020 -0.500
> anova(final.mod)
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
factor(congruity) 9221.9 9221.9 1 46129 49.518 1.993e-12 ***
factor(laterality) 3511.5 1755.7 2 46095 9.428 8.062e-05 ***
factor(anteriority) 23814.0 23814.0 1 46095 127.873 < 2.2e-16 ***
factor(congruity):factor(laterality) 680.3 340.1 2 46095 1.826 0.16101
factor(congruity):factor(anteriority) 411.5 411.5 1 46095 2.210 0.13714
factor(laterality):factor(anteriority) 1497.4 748.7 2 46095 4.020 0.01796 *
factor(congruity):factor(laterality):factor(anteriority) 8.6 4.3 2 46095 0.023 0.97713
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1