理解混合效应模型：随机效应与实证分析

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"这篇文章探讨了在确认性假设测试中如何使用线性混合效应模型（Linear Mixed-Effects Models, LMEMs）以及随机效应结构对分析结果的影响。文章指出，许多研究者可能没有充分理解随机效应结构如何影响分析的可推广性。作者主张，使用LMEMs的研究者应至少遵循多年来的标准，通过理论论证和蒙特卡洛模拟来强调这一点。" 混合效应模型是一种统计建模方法，它结合了固定效应和随机效应，用于处理具有嵌套或等级结构的数据，如时间序列数据、重复测量数据或者具有多个层次的数据（如学校、班级、学生等）。这种模型特别适合处理非独立观察，因为它允许在不同组间存在差异，同时考虑组内的相关性。随机效应的数据结构在混合效应模型中扮演关键角色。随机效应通常用于表示那些无法完全观测到或不可控制的变量对结果的影响。例如，在教育研究中，班级可能是随机效应，因为每个班级的教师教学风格可能有所不同，但无法完全测量。随机效应模型能够捕捉这种未观测到的变异性，提高模型的适应性和解释力。文章强调了在确认性假设测试中采用混合效应模型时，随机效应结构的重要性。研究人员应该认识到，随机效应的选择不仅影响模型的拟合，还直接影响结果的推断和统计功效。保持随机效应结构的最大化，意味着包括所有可能的随机效应，以确保模型的全面性和准确性。论文通过理论分析和蒙特卡洛模拟，展示了随机效应结构如何影响模型的统计性质和推断结论。蒙特卡洛模拟是一种统计方法，通过大量的随机抽样来估计模型的性能。这种方法可以帮助研究人员理解在各种条件下模型的行为，从而更好地设计和解释他们的分析。关键词：线性混合效应模型、泛化、统计学、蒙特卡洛模拟。这些关键词突出了文章的核心内容，即使用LMEMs进行统计分析时，如何处理随机效应以提高结果的可靠性和广泛适用性，并通过计算方法验证这些理论观点。这篇文章提供了关于如何在心理学语言学和其他相关领域中正确使用混合效应模型的见解，强调了随机效应的适当选择对于确认性假设测试的重要性，并通过理论和实际模拟提供了深入的理解。

where X

is a predictor variable

taking on the value of 0 or

1 depending on whether item i is of type A or B respectively,

and e

 N(0,

) indicates that the trial-level error is nor-

mally distributed with mean 0 and variance

. In the pop-

ulation, participants respond to items of type B 40 ms faster

than items of type A. Under this ﬁrst model, we assume that

each of the 16 observations provides the same evidence for

or against the treatment effect regardless of whether or not

any other observations have already been taken into ac-

count. Performing an unpaired t-test on these data would

implicitly assume this (incorrect) generative model.

Model (1) is not a mixed-effects model because we have

not deﬁned any sources of clustering in our data; all obser-

vations are conditionally independent given a choice of

intercept, treatment effect, and noise level. But experience

tells us that different subjects are likely to have different

overall response latencies, breaking conditional indepen-

dence between trials for a given subject. We can expand

our model to account for this by including a new offset

term S

, the deviation from b

for subject s. The expanded

model is now

¼ b

þ S

þ b

þ e

;

 Nð0;

Þ;

 Nð0;

Þ:

ð2Þ

These offsets increase the model’s expressivity by allowing

predictions for each subject to shift upward or downward

by a ﬁxed amount (Fig. 1b). Our use of Latin letters for this

term is a reminder that S

is a special type of effect which

is different from the bs—indeed, we now have a ‘‘mixed-ef-

fects’’ model: parameters b

and b

are ﬁxed effects (effects

that are assumed to be constant from one experiment to

another), while the speciﬁc composition of subject levels

for a given experiment is assumed to be a random subset

of the levels in the underlying populations (another instan-

tiation of the same experiment would have a different

composition of subjects, and therefore different realiza-

tions of the S

effects). The S

effects are therefore random

effects; speciﬁcally, they are random intercepts, as they al-

low the intercept term to vary across subjects. Our primary

goal is to produce a model which will generalize to the

population from which these subjects are randomly drawn,

rather than describing the speciﬁc S

values for this sam-

ple. Therefore, instead of estimating the individual S

ef-

fects, the model-ﬁtting algorithm estimates the

population distribution from which the S

effects were

drawn. This requires assumptions about this distribution;

we follow the common assumption that it is normal, with

a mean of 0 and a variance of

; here

is a random effect

parameter, and is denoted by a Greek symbol because, like

the bs, it refers to the population and not to the sample.

Note that the variation on the intercepts is not con-

founded with our effect of primary theoretical interest

): for each subject, it moves the means for both condi-

tions up or down by a ﬁxed amount. Accounting for this

variation will typically decrease the residual error and thus

increase the sensitivity of the test of b

. Fitting Model (2) is

thus analogous to analyzing the raw, unaggregated re-

sponse data using a repeated-measures ANOVA with SS

sub-

jects

subtracted from the residual SS

error

term. One could see

that this analysis is wrong by observing that the denomi-

nator degrees of freedom for the F statistic (i.e., corre-

sponding to MS

error

) would be greater than the number of

subjects (see Online Appendix for further discussion and

demonstration).

Although Model (2) is clearly preferable to Model (1),it

does not capture all the possible by-subject dependencies

in the sample; experience also tells us that subjects often

vary not only in their overall response latencies but also

in the nature of their response to word type. In the present

hypothetical case, Subject 3 shows a total effect of

134 ms, which is 94 ms larger than the average effect in

the population of 40 ms. We have multiple observations

per combination of subject and word type, so this variabil-

ity in the population will also create clustering in the sam-

ple. The S

do not capture this variability because they

only allow subjects to vary around b

. What we need in

addition are random slopes to allow subjects to vary with

respect to b

, our treatment effect. To account for this var-

iation, we introduce a random slope term S

with variance

, yielding

¼ b

þ S

þðb

þ S

ÞX

þ e

;

ðS

; S

ÞN 0;

"# !

;

 Nð0;

Þ:

ð3Þ

This is now a mixed-effects model with by-subject random

intercepts and random slopes. Note that the inclusion of the

by-subject random slope causes the predictions for condi-

tion B to shift by a ﬁxed amount for each subject (Fig. 1c),

improving predictions for words of type B. The slope offset

captures how much Subject s’s effect deviates from the

population treatment effect b

. Again, we do not want our

analysis to commit to particular S

effects, and so, rather

than estimating these values directly, we estimate

the by-subject variance in treatment effect. But note that

now we have two random effects for each subject s, and

these two effects can exhibit a correlation (expressed by

). For example, subjects who do not read carefully might

not only respond faster than the typical subject (and have a

negative S

), but might also show less sensitivity to the

word type manipulation (and have a more positive S

). In-

deed, such a negative correlation, where we would have

< 0, is suggested in our hypothetical data (Fig. 1): S1

and S3 are slow responders who show clear treatment ef-

fects, whereas S2 and S4 are fast responders who are

hardly susceptible to the word type manipulation. In the

most general case, we should not treat these effects as

coming from independent univariate distributions, but in-

stead should treat S

and S

as being jointly drawn from a

For expository purposes, we use a treatment coding scheme (0 or 1) for

the predictor variable. Alternatively, the models in this section could be

expressed in the style more common to traditional ANOVA pedagogy,

where ﬁxed and random effects represent deviations from a grand mean.

This model can be ﬁt by using ‘‘deviation coding’’ for the predictor variable

(.5 and .5 instead of 0 and 1). For higher-order designs, treatment and

deviation coding schemes will lead to different interpretations for lower-

order effects (simple effects for contrast coding and main effects for

deviation coding).

D.J. Barr et al. / Journal of Memory and Language 68 (2013) 255–278

259

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