GENERAL LINEAR MODELS

Oneway ANOVA, GLM Univariate (n-way ANOVA,

ANCOVA)

BASICS 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

BASICSFactor A (industry)Level 1 (manufact)

Factor A (industry)Level 2 (trade)

Factor B (size)Level 1 (small)

Cell

Factor B (size)Level 2 (medium)Factor B (size)Level 3 (large)

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

Balanced design: each cell has equal number of observations

ASSUMPTIONS 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

STEPS OF INTERPRETATION 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

Oneway ANOVA 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

Contrast Coefficients

1 0 -1 0,5 ,5 -1 0

Contrast12

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SAS: oneway ANOVA

SAS: oneway ANOVA

BF or Levene, H0: group variances are equal

Use this instead of F if variances are not equal

SAS: oneway ANOVAPost hoc -tests

SAS: oneway ANOVA

SAS: oneway ANOVA

MODEL FIT

Source DFSum of

Squares Mean Square F Value Pr > FModel 3 298.3992640 99.4664213 13.41 <.0001Error 68 504.4139305 7.4178519

Corrected Total 71 802.8131944

R-Square Coeff Var Root MSE deathrate Mean0.371692 34.10981 2.723573 7.984722

Class Level Information

Class Levels Valuesclass_popgrowth 4 1 2 3 4

EQUALITY OF VARIANCESLevene's Test for Homogeneity of deathrate VarianceANOVA of Squared Deviations from Group Means

Source DFSum of

SquaresMean

Square F Value Pr > Fclass_popgrowth 3 3004.3 1001.4 4.23 0.0084

Error 68 16110.6 236.9

Welch's ANOVA for deathrate

Source DF F Value Pr > Fclass_popgrowth 3.0000 13.00 <.0001

Error 18.9519

GROUP MEANS

Level ofclass_popgrowth N

deathrate

Mean Std Dev1 27 10.5666667 3.01457996

2 22 6.9272727 1.60064922

3 17 5.9705882 1.87941637

4 6 5.9500000 5.61809576

POST HOC TESTComparisons significant at the 0.05 level are indicated by ***.

class_popgrowthComparison

DifferenceBetween

MeansSimultaneous 95% Confidence Limits

1 - 2 3.6394 1.5135 5.7653 ***1 - 3 4.5961 2.3044 6.8877 ***1 - 4 4.6167 1.2760 7.9573 ***2 - 1 -3.6394 -5.7653 -1.5135 ***2 - 3 0.9567 -1.4335 3.34682 - 4 0.9773 -2.4317 4.38623 - 1 -4.5961 -6.8877 -2.3044 ***3 - 2 -0.9567 -3.3468 1.43353 - 4 0.0206 -3.4942 3.53534 - 1 -4.6167 -7.9573 -1.2760 ***4 - 2 -0.9773 -4.3862 2.43174 - 3 -0.0206 -3.5353 3.4942

BOXPLOTS

Multiway ANOVA, GLM A continuous dependent variable y, two or

more categorical independent variables (factorial design)

ANCOVA, if there are continuous independents (covariates)

main effects and interaction effects can be modeled

fixed factor, if all groups are present and random factor, if only some groups are randomly represented in the data

Eta squared = SSK/SST expresses how many % of the variance in y is explained by 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

mean of profitability

0

10

20

30

40

small medium large

manufact

trade

INTERACTIONSOrdinal 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)

mean of profitability

0

10

20

30

40

50

small medium large

manufact

trade

mean of profitability

0

10

20

30

40

50

small medium large

manufact

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 – linear models

Effects to be estimated

Interaction here, first select both variables, then click Cross

Sums of squares

Other options, defaults ok

Post hoc-tests

Plots

SAS - codePROC GLM DATA=libname.datafilename

PLOTS(ONLY)=DIAGNOSTICS(UNPACK)PLOTS(ONLY)=RESIDUALSPLOTS(ONLY)=INTPLOT;CLASS Elinkaari Perheyr;MODEL growthorient= ln_hlo Elinkaari Perheyr Elinkaari*Perheyr/SS3SOLUTIONSINGULAR=1E-07EFFECTSIZE;LSMEANS Elinkaari Perheyr Elinkaari*Perheyr / PDIFF ADJUST=BON ;RUN;

QUIT;

Model significance and fitClass Level Information

Class Levels ValuesElinkaari phase 3 2 3 4Perheyr family 2 0 1

Number of Observations Read 181Number of Observations Used 132

Source DFSum of

Squares Mean Square F Value Pr > FModel 6 13.03085542 2.17180924 3.59 0.0026Error 125 75.69810081 0.60558481Corrected Total 131 88.72895623

R-Square Coeff Var Root MSE growthorient Mean0.146861 21.79382 0.778193 3.570707

Significance of predictorsSource DF Type III SS Mean Square F Value Pr > Fln_hlo employees 1 2.88693851 2.88693851 4.77 0.0309Elinkaari phase 2 9.52176337 4.76088169 7.86 0.0006Perheyr family 1 0.28960870 0.28960870 0.48 0.4905Elinkaari*Perheyr

Phase*Family

2 1.99071120 0.99535560 1.64 0.1974

EFFECT SIZE OF PREDICTORS

Source

Total Variation Accounted For Partial Variation Accounted For

Semipartial Eta-Square

Semipartial Omega-Square

Conservative95% Confidence Li

mitsPartial Eta-

Square

Partial Omega-Square

95% Confidence Limits

ln_hlo 0.0325 0.0255 0.0000 0.1112 0.0367 0.0277 0.0000

0.1158

Elinkaari 0.1073 0.0930 0.0219 0.2056 0.1117 0.0942 0.0225

0.2073

Perheyr 0.0033 -0.0035 0.0000 0.0488 0.0038 -0.0040 0.0000

0.0503

Elinkaari*Perheyr

0.0224 0.0087 0.0000 0.0842 0.0256 0.0097 0.0000

0.0887

Parameter estimatesParameter Estimate

Standard Error t Value Pr > |t|

Intercept 3.196306815 B 0.49826714 6.41 <.0001ln_hlo employees 0.161079578 0.07377500 2.18 0.0309Elinkaari 2 growth 0.372704251 B 0.49030119 0.76 0.4486Elinkaari 3 mature -0.041166136 B 0.46224369 -0.09 0.9292Elinkaari 4 decline 0.000000000 B . . .Perheyr 0 non family -0.862973482 B 0.92404272 -0.93 0.3522Perheyr 1 family 0.000000000 B . . .Elinkaari*Perheyr 2 0 1.250588328 B 0.98491805 1.27 0.2065Elinkaari*Perheyr 2 1 0.000000000 B . . .Elinkaari*Perheyr 3 0 0.654885600 B 0.94241380 0.69 0.4884Elinkaari*Perheyr 3 1 0.000000000 B . . .Elinkaari*Perheyr 4 0 0.000000000 B . . .Elinkaari*Perheyr 4 1 0.000000000 B . . .

38

Prediction for 6 cells Elinkaari=2 & perheyr=0 (growth phase, non family)

Growth = 3.20 + 0.16*ln_hlo + 0.37 – 0.86 + 1.25= 3.96 + 0.16*ln_hlo

Elinkaari=3 & perheyr=0 (mature phase, non family)Growth = 3.20 + 0.16*ln_hlo – 0.04 – 0.86 + 0.65

= 2.95 + 0.16*ln_hlo Elinkaari=4 & perheyr=0 (decline phase, non family)

Growth = 3.20 + 0.16*ln_hlo + 0.00 – 0.86 + 0.00= 2.34 + 0.16*ln_hlo

Elinkaari=2 & perheyr=1 (growth phase, family)Growth = 3.20 + 0.16*ln_hlo + 0.37 + 0.00 + 0.00

= 3.57 + 0.16*ln_hlo Elinkaari=3 & perheyr=1 (mature phase, family)

Growth = 3.20 + 0.16*ln_hlo - 0.04 + 0.00 + 0.00= 3.16 + 0.16*ln_hlo

Elinkaari=4 & perheyr=1 (decline phase, family)Growth = 3.20 + 0.16*ln_hlo + 0.00 + 0.00 + 0.00

= 3.20 + 0.16*ln_hlo

39

Parameter estimatesThe X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable.

This warning always occurs if you have categorical independent variables in the model, SAS can however estimate the coefficients

Homoskedasticity

Outlier diagnostics

Residual distribution

Model fit

Influence diagnostics

Residual vs. covariate

Significance of group differences, main effects

Elinkaari

phasegrowthorient

LSMEANLSMEAN

Number2 growth 4.14643211 13 mature 3.43471035 24 decline 3.14843369 3

Least Squares Means for effect ElinkaariPr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: growthorienti/j 1 2 31 0.0006 0.12252 0.0006 1.00003 0.1225 1.0000

Perheyr

Familygrowthorient

LSMEAN

H0:LSMean1=LSMean2

Pr > |t|0 3.46261763 0.4905

1 3.69043314

Significance of group differences, interaction

Phase Familygrowthorient

LSMEANLSMEAN

Number2 growth 0 4.34023953 12 1 3.95262468 23 mature 0 3.33066641 33 1 3.53875430 44 decline 0 2.71694695 54 1 3.57992043 6

Least Squares Means for effect Elinkaari*PerheyrPr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: growthorienti/j 1 2 3 4 5 61 1.000

00.016

10.105

20.847

41.0000

2 1.0000

0.1040

0.8177

1.0000

1.0000

3 0.0161

0.1040

1.0000

1.0000

1.0000

4 0.1052

0.8177

1.0000

1.0000

1.0000

5 0.8474

1.0000

1.0000

1.0000

1.0000

6 1.0000

1.0000

1.0000

1.0000

1.0000

Non-family firms in growth phase differ from non-family firms in mature phase

REPORTING GLM Model fit: F + df + p and R Square Nature and significance of effects:

parameter estimates B+s.e.+t+p and F+p

estimated group means (means plot)

post hoc test results

Means plot

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1.5

2

2.5

3

3.5

4

4.5

5

perheyrei-perheyr

kasv

uhak

uisu

us

Employees at its mean value (20)