1 / 103

2.54k likes | 5.24k Views

Linear Mixed Models: An Introduction. Patrick J. Rosopa, Ph.D. University of Central Florida. Outline. The Need for Linear Mixed Models General Linear Models versus Linear Mixed Models A Quick Note Regarding Software Types of Effects Some Common Models Estimation of Parameters

Download Presentation
## Linear Mixed Models: An Introduction

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Linear Mixed Models:An Introduction**Patrick J. Rosopa, Ph.D. University of Central Florida**Outline**• The Need for Linear Mixed Models • General Linear Models versus Linear Mixed Models • A Quick Note Regarding Software • Types of Effects • Some Common Models • Estimation of Parameters • Goodness-of-Fit Indices • Statistical Inferences • Covariance Structures • Linear Mixed Models in SPSS • Some Examples Using SPSS**Grouped Data is Common**• In the social and behavioral sciences, including industrial and organizational psychology, grouped or clustered data is quite common (Hox, 2002; Raudenbush & Bryk, 2002). • Team members nested within teams. • Students nested within schools. • Employees nested within departments & departments within organizations. • Interviewees nested within interviewer. • Randomized block designs. • In longitudinal and growth curve research, several repeated measurements are nested within individuals. • In meta-analysis, participants are nested within different studies.**Person 1**Score time1 Score time2 Score time3**To Aggregate or Not to Aggregate?**• In some situations, all variables are aggregated to the “higher” level (e.g., team, school). • In some situations, all variables are disaggregated to the “lower” level (e.g., team member, student). • In both, ordinary least squares (OLS) “fixed effects” general linear models are frequently used. (regression or ANOVA of some kind)**To Aggregate or Not to Aggregate?**• In some situations, all variables are aggregated to the “higher” level (e.g., team, school).**Compute Mean**Compute Mean**To Aggregate or Not to Aggregate?**• In some situations, all variables are aggregated to the “higher” level (e.g., team, school). • Result: Within-group information is lost, smaller sample size, and loss of statistical power.**To Aggregate or Not to Aggregate?**• In some situations, all variables are aggregated to the “higher” level (e.g., team, school). • Result: Within-group information is lost, smaller sample size, and loss of statistical power. • In some situations, all variables are disaggregated to the “lower” level (e.g., team member, student).**To Aggregate or Not to Aggregate?**• In some situations, all variables are aggregated to the “higher” level (e.g., team, school). • Result: Within-group information is lost, smaller sample size, and loss of statistical power. • In some situations, all variables are disaggregated to the “lower” level (e.g., team member, student). • Result: Observations are treated as if they were independent when they really aren’t, resulting in overly optimistic (i.e., smaller) standard errors and increased Type I error rate. (unjustified increase in power)**Linear Mixed Models as a Solution**• Linear mixed models used to account for: • the lack of independence, i.e., correlation, among observations • the unequal errors, i.e., heteroscedasticity, across observations (non-constant variance across groups) • both lack of independence and heteroscedasticity across observations • Additional advantages: • It can handle missing values and will not drop cases from the analysis (unlike standard repeated measures ANOVA). • Time can be treated flexibly (Singer & Willett, 2003). (e.g., One person measured on the dependent variable at Time 1, Time 4, & Time 6. Another person measured at Time 1, Time 3, Time 5, & Time 6.). • When used in Generalizability Theory, the common estimation methods (i.e., ML & REML, to be discussed later) will not result in negative variance components.**Linear Mixed Models as a Solution**• “The Linear Mixed Models procedure expands the general linear model so that the error terms and random effects are permitted to exhibit correlated (non independent) and non-constant variability (heteroscedasticity). The linear mixed model, therefore, provides the flexibility to model not only the mean of a response variable, but its covariance structure as well.” (SPSS, 2006)**General Linear Models vs.Linear Mixed Models**• How does a traditional general linear model (GLM) differ from a linear mixed model (LMM)? • Some traditional GLMs: • Analyses involving the mean of a single group (1 sample t test) • Analyses involving the means of 2 independent groups (1 categorical IV) • Analyses involving the means of > 2 independent groups (1 categorical IV) • Factorial designs (2 or more completely crossed categorical IVs) • Multiple regression (continuous IVs) • ANCOVA (continuous and categorical IVs), etc. y = (Fixed Effects) + random error**General Linear Models vs.Linear Mixed Models**GLM: y = (Fixed Effects) + random error LMM: y = (Fixed Effects) + (Random Effects) + random error • When there are no random effects, we have the usual fixed effects general linear model (Pinheiro & Bates, 2000; Searle, Casella, & McCulloch, 1992). This model is covered in detail by numerous researchers (Cohen, Cohen, West, & Aiken, 2003; Fox, 1997; Pedhazur, 1982).**Some Terminology**• Linear mixed models, hierarchical linear models (psych), multilevel models (edu, and psych), and random coefficient models are used interchangeably. • Note that these are computed using a model that incorporates fixed and/or random effects; thus, major software packages use “mixed” or “mixed effects.” • SPSS procedure: MIXED. • SAS procedure: PROC MIXED. • S-PLUS function: lme. (linear mixed effects) • R function: lme.**General-Purpose Statistical Software**• Able to fit linear mixed models.**MLwiN**Specialized Statistical Software • Developed specifically for linear mixed models.**Fixed vs. Random Effects**• Fixed effects: used to model the mean or predicted value of Y. The levels or values of the predictor are not sampled from a larger population. • Sex • Ethnicity • Fixed effects take the form of the usual regression coefficients, e.g., standardized or unstandardized. (slopes and intercepts) • In the literature on hierarchical linear modeling, random coefficient modeling, and multilevel modeling, the fixed effects are typically denoted by gammas (γ) instead of betas (β). • As in general linear models, you will be able to obtain the correlation and/or covariance matrix among the fixed effects coefficients in SPSS.**Fixed vs. Random Effects**• Random effects: used to account for the variance-covariance of DV after the fixed effects have been accounted for (of Y). The levels or values observed have been sampled from a larger population of possible levels or values. • Teams • Schools • Persons (in longitudinal studies) • Clinics (e.g., selecting a representative sample of clinics to test for the effectiveness of a new procedure) • Blocks (in a randomized block design)**Fixed vs. Random Effects**• In the literature on hierarchical linear modeling, random coefficient modeling, and multilevel modeling, the random effects are denoted by u. • Instead of e for residuals or ε for errors, r is used. • Creating an interaction term that involves both a fixed effect AND a random effect results in a random effect term. • e.g., the interaction between a fixed effect term (sex) and random effect term (Team ID) results in a random effect term (sex × Team ID). • In statistical software, you will be able to obtain the correlation and/or covariance matrix among the random effects. • e.g., If you have 2 random effects, there will be a 2 × 2 covariance matrix for the random effects. This contains the estimated variance components.**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). • Most basic model you can fit for a linear mixed model • No predictors**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). Level 1: Level 2: LMM:**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). Mean in jth School/Team/Cluster Level 1: Level 2: LMM:**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). Level 1: Grand Mean, that is, . . . Level 2: LMM:**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). Level 1: Level 2: LMM: Fixed Effect (Intercept)**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). Level 1: Level 2: LMM: Random Effect for jth School/Team/Cluster**Null Model**• Also called the fully unconditional model (Raudenbush & Bryk, 2002). • In the experimental design literature, it is known as a one-way random effects ANOVA (Kirk, 1995). Level 1: Level 2: LMM: Residual**Null Model**• From this model, we can estimate the intraclass correlation (ICC(1,1)), Shrout & Fleiss, 1979) or cluster effect (Raudenbush & Bryk, 2002; Snijders & Bosker, 2003). • A large value suggests that differences between schools/teams/clusters are larger than differences within them. Inappropriate to treat observations within the same school/team/cluster as independent. LMM:**Null Model**• We can also use the variance component from the null model to determine whether we can explain additional variability between schools/teams/clusters by including additional variables. Proportion of Explained Variance**Adding a Fixed Effect from Level 2**• Adds a predictor • First example had one fixed effect (i.e., the intercept) and one random effect due to the school/team/cluster. • We could add a fixed effect from Level 2 (e.g., a predictor on which we have the mean within the group, a team characteristic).**Adding a Fixed Effect from Level 2**• First example had one fixed effect (i.e., the intercept) and one random effect due to the school/team/cluster. • We could add a fixed effect from Level 2 (e.g., a predictor on which we have the mean within the group, a team characteristic). Level 1: Level 2: LMM: Fixed Effect (Intercept) Fixed Effect (Slope)**Adding a Fixed Effect from Level 2**• First example had one fixed effect (i.e., the intercept) and one random effect due to the school/team/cluster. • We could add a fixed effect from Level 2 (e.g., a predictor on which we have the mean within the group, a team characteristic). Level 1: Level 2: LMM: Random Effect for jth School/Team/Cluster**Adding a Fixed Effect from Level 2**• First example had one fixed effect (i.e., the intercept) and one random effect due to the school/team/cluster. • We could add a fixed effect from Level 2 (e.g., a predictor on which we have the mean within the group, a team characteristic). Level 1: Level 2: LMM: Residual**Adding a Fixed Effect from Level 1**• First example had one fixed effect (i.e., the intercept) and one random effect due to the school/team/cluster. • We could add a fixed effect from Level 1 (e.g., a predictor on which we have values that vary within a group, but are not sampled from a larger population).**Adding a Fixed Effect from Level 1**Level 1: Level 2: LMM:**Adding a Fixed Effect from Level 1**Level 1: Level 2: LMM: Fixed Effect (Intercept) Fixed Effect (Slope)**Adding a Fixed Effect from Level 1**Level 1: Level 2: LMM: Random Effect for jth School/Team/Cluster**Adding a Fixed Effect from Level 1**Level 1: Level 2: LMM: Residual**Adding a Fixed Effect Covariate from Level 1**• First example had one fixed effect (i.e., the intercept) and one random effect due to the school/team/cluster. • Add a fixed effect covariate from Level 1 (e.g., cognitive ability scores which have values that vary within a group, but are not sampled from a larger population). • This is a one-way ANCOVA with a random effect for the school/team/cluster.**Adding a Fixed Effect Covariate from Level 1**Level 1: Level 2: LMM:

More Related