General linear models
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GENERAL LINEAR MODELS. Oneway ANOVA, GLM Univariate (n-way ANOVA, ANCOVA) . Dependent variable is continuous Independent variables are nominal, categorical (factor, CLASS) or continuous (covariate)

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GENERAL LINEAR MODELS

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GENERAL LINEAR MODELS

Oneway ANOVA,

GLM Univariate (n-way ANOVA, ANCOVA)


Dependent variable is continuous

Independent variables are nominal, categorical (factor, CLASS) or continuous (covariate)

Are the group means of the dependent variable different across groups defined by the independents

Main effects, interactions and nested effects

Often used for testing hypotheses with experimental data

BASICS


BASICS

3 X 2 full factorial design (full: each cell has observations)

Balanced design: each cell has equal number of observations


Enough observations in each group? (n >20)

Independence of observations

Similarity of variance-covariance matrices (no problem if largest group variance < 1.5*smallest group variance, 4* if balanced design)

Normality

Linearity

No outlier-observations

ASSUMPTIONS


Model significance?

F-test and R square

Welch, if unequal group variances (this can be tested using Levene or Brown-Forsythe test)

Significance of effects? (F-test and partial eta squared)

Which group differences are significant? Post hoc or contrast tests

What are the group differences like? Estimated marginal means for groups

STEPS OF INTERPRETATION


A continuous dependent variable (y) and one categorical independent variable (x), with min. 3 categories, k= number of categories

assumptions: y normally distributed with equal variance in each x category

H0: mean of y is the same in all x categories

Variance of y is divided into two components: within groups (error) and between groups (model, treatment)

Test statistic= between mean square / within mean square follows F-distribution with k-1, n-k degrees of freedom

F-test can be replaced by Welch if variances are unequal

Oneway ANOVA


If the F test is significant, you can use post hoc tests for pairwise comparison of means across the groups

Alternatively (in experiments) you can define contrasts ex ante

Oneway ANOVA


SAS: oneway ANOVA


SAS: oneway ANOVA

Use this instead of F if variances are not equal

BF or Levene, H0: group variances are equal


SAS: oneway ANOVA

Post hoc -tests


SAS: oneway ANOVA


SAS: oneway ANOVA


MODEL FIT


EQUALITY OF VARIANCES


GROUP MEANS


POST HOC TEST


BOXPLOTS


Multiway ANOVA, GLM

  • A continuousdependentvariable y, twoormorecategoricalindependentvariables (factorial design)

  • ANCOVA, iftherearecontinuousindependents (covariates)

  • main effects and interactioneffectscanbemodeled

  • fixedfactor, ifallgroupsarepresent and randomfactor, ifonlysomegroupsarerandomlyrepresented in the data

  • Eta squared = SSK/SST expresseshowmany % of the variance in y is explainedby x (not in EG! SAS code: model y = x1 x2 / ss3 EFFECTSIZE;)


INTERACTION EFFECT

  • Synergy of two factors, the effect of one factor is different in the groups of the other factor

  • Crossing effect = interaction effect

  • Ordinal (lines in means plot have different slopes, but do not cross)

  • Disordinal (lines cross in the means plot)


NO INTERACTION

Size and industry both have a significant main effect

No interaction, homogeneity of slopes


INTERACTIONS

Ordinal interaction (the effect of size is stronger in manufacturing than in trade)

Dis-ordinal interaction (the effect of size has a different sign in manufacturing and trade)


NESTED EFFECTS

  • Nested effect B(A) ”B nested within A”

  • size (industry): the effect of size is estimated separately for each industry group

  • Difference between nested and interaction effect is that the main effect of B (size) is not included

  • The slope of B (size) is different in each category of A (industry)


ESTIMATED GROUP MEANS

  • Estimated marginal means or LS (least squares) means

  • Predicted group means are calculated using the estimated model coefficients

  • The effects of other independent variables are controlled for

  • Is not equal to the group means from the sample


SUM OF SQUARES

  • Type I SS does not control for the effects of other independent variables which are specified later into the model

  • Type II SS controls for the effects of all other independents

  • Types III and IV SS are better in unbalanced designs, IV if there are empty cells


POST HOC TESTS

  • Multiple comparison procedures, mean separation tests

  • The idea is to avoid the risk of Type I error which results from doing many pairwise tests, each at 5% risk level

  • E.g. Bonferroni, Scheffe, Sidak,…

  • Tukey-Kramer is most powerful

  • H0: equal group means -> rejection means that group means are not equal, but failure to reject does not necessarily mean that they are equal (small sample size -> low power -> failure to reject the null)


ANCOVA

  • The model includes a covariate (= continuous independent variable, often one whose effect you want to control for)

  • Regress y on the covariate -> then ANOVA with factors explaining the residual

  • The relationship between covariate and y must be linear, and the slope is assumed to be the same at all factor levels

  • The covariate and factor should not be too much related to each other

  • Do not include too many covariates, max 0.1*n – (k-1)


SAS: analyze – ANOVA – linearmodels


Effects to beestimated

Interactionhere, firstselectbothvariables, thenclickCross


Sums of squares


Otheroptions, defaults ok


Post hoc-tests


Plots


SAS - code

PROC GLM DATA=libname.datafilename

PLOTS(ONLY)=DIAGNOSTICS(UNPACK)

PLOTS(ONLY)=RESIDUALS

PLOTS(ONLY)=INTPLOT

;

CLASS Elinkaari Perheyr;

MODEL growthorient=ln_hlo Elinkaari PerheyrElinkaari*Perheyr

/

SS3

SOLUTION

SINGULAR=1E-07

EFFECTSIZE

;

LSMEANS Elinkaari PerheyrElinkaari*Perheyr / PDIFF ADJUST=BON ;

RUN;

QUIT;


Modelsignificance and fit


Significance of predictors


EFFECT SIZE OF PREDICTORS


Parameterestimates


Prediction for 6 cells

  • Elinkaari=2 & perheyr=0 (growthphase, nonfamily)

    Growth= 3.20 + 0.16*ln_hlo + 0.37 – 0.86 + 1.25

    = 3.96 + 0.16*ln_hlo

  • Elinkaari=3 & perheyr=0 (maturephase, nonfamily)

    Growth = 3.20 + 0.16*ln_hlo – 0.04 – 0.86 + 0.65

    = 2.95 + 0.16*ln_hlo

  • Elinkaari=4 & perheyr=0 (declinephase, nonfamily)

    Growth = 3.20 + 0.16*ln_hlo + 0.00 – 0.86 + 0.00

    = 2.34 + 0.16*ln_hlo

  • Elinkaari=2 & perheyr=1 (growthphase, family)

    Growth = 3.20 + 0.16*ln_hlo + 0.37 + 0.00 + 0.00

    = 3.57 + 0.16*ln_hlo

  • Elinkaari=3 & perheyr=1 (maturephase, family)

    Growth = 3.20 + 0.16*ln_hlo - 0.04 + 0.00 + 0.00

    = 3.16 + 0.16*ln_hlo

  • Elinkaari=4 & perheyr=1 (declinephase, family)

    Growth = 3.20 + 0.16*ln_hlo + 0.00 + 0.00 + 0.00

    = 3.20 + 0.16*ln_hlo


Parameterestimates


Homoskedasticity


Outlierdiagnostics


Residualdistribution


Modelfit


Influencediagnostics


Residual vs. covariate


Significance of groupdifferences, main effects


Significance of groupdifferences, interaction

Non-familyfirms in growthphasedifferfromnon-familyfirms in maturephase


REPORTING GLM

  • Modelfit: F + df + p and R Square

  • Nature and significance of effects: parameterestimatesB+s.e.+t+p and F+p

  • estimatedgroupmeans (meansplot)

  • posthoctestresults


Meansplot

Employees at itsmeanvalue (20)


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