## Statistics & Methods Centre - (M)AN(C)OVA models

- Analysis-of-variance (ANOVA) models belong to the family of general linear models with one or more response (dependent) variables
and at least one (multi-)categorical predictor or factor. Factorial analysis-of-variance models are models with more than one categorial
predictor. Multivariate ANOVA models have several response variables, ANCOVA
models have at least one numerical and one categorical predictor. In Repeated measures ANOVA subjects are measured on
more than once on the same response variable, a situation which typically arises in longitudinal studies.

Detailed information on the tests is contained in the books mentioned.

- One-way analysis of variance
- Factorial analysis of variance
- Multivariate analysis of variance
- Analysis of covariance
- Repeated-measures analysis of variance

- One-way analysis-of-variance models - ANOVA
**Basic reading**- Moore & McCabe, Chapter 12: One-way analysis of variance;

Field, Chapter 10: Comparing several means: ANOVA (GLM 1) *Effect sizes:*Field, Chapter 10, section: 10.5**Advanced reading**- Field, Chapter 10, 12, 13; later sections of these chapters
**Software***Means and st. deviations:*- SPSS => Analyze => Univariate => Options => Descriptive
*One-way ANOVA:*- SPSS => Analyze => Compare Means => One-Way ANOVA
*Effect sizes:*- SPSS => Analyze => General Linear Model => Options => Estimate effect size
**Reporting analysis of variance in publications**- The results of an
*One-way ANOVA*are reported with between brackets the appropriate degrees-of-freedom of first the between sum-of-squares and then the within sum-of-squares, the value of the*F*-statistic in two decimals (*F*(*df*_{between},*df*_{within}), followed by the descriptive level of significance (*p*). Also report effect sizes (mostly partial eta-squared), and if appropriate post-hoc comparisons.

**Example:**A statistically significant main effect for treatments was observed,*F*(2,145) = 5.43,*p*< .01. (*M*_{exp1}= 7.5,*SD*= 1.00,*N*= 30;*M*_{exp2}= 8.5,*SD*= 1.00,*N*= 30;*M*_{control}= 5.5,*SD*= 1.00,*N*= 30). The partial eta-squared (η^{2}= .06) was of medium size.

Suggested norms for partial eta-squared: small = 0.01; medium = 0.06; large = 0.14. **Reporting - Advice**: Field (2005), Section 8.6; Pallant (2007), p. 248.**Power:**Power calculator**Effect size:**Calculated in SPSS via the Generalized Linear Models program; see also Field (2005), section 8.5.

- Factorial analysis-of-variance models - ANOVA
**Basic reading**- Moore & McCabe, Chapter 13: Two-way analysis of variance;

Field, Chapter 10: Factorial ANOVA (GLM 3) *Effect sizes:*Field, Chapter xx, section:**Advanced reading****Software****Reporting - Advice**

The results for a*Factorial ANOVA*can best be reported in a table.

**Example:**

Table 8

*Response variable: Aggresion of children during play with war toys*(*N*=146)Source *SS**df**MS**F**p**η*^{2}Family composition 7.18 2 3.59 4.46 .013 .060 Attitude towards war toys 33.93 1 33.93 42.12 <.001 .231 Interaction 5.60 2 2.80 3.47 0.34 0.47 Within (Residual) 112.79 140 .81 Total 165.56 145 **Power:****Effect size:**

- Multivariate analysis-of-variance models - MANOVA
**Basic reading**Model Textbook *Multivariate ANOVA:*Field, Chapter xx: xxxxxxx *Effect sizes:*Field, Chapter xx, section: **Advanced reading****Software****Reporting - Advice****Power:****Effect size:**

- Analysis of covariance - ANCOVA
**Basic reading**- Moore & McCabe, Chapter 13: Two-way analysis of variance;

Field, Chapter 10: Factorial ANOVA (GLM 3) *Effect sizes:*Field, Chapter xx, section:**Advanced reading****Software****Reporting - Advice****Power:****Effect size:**

- Repeated-measures analysis of variance - Repeated Measures ANOVA
**Basic reading**- Field, Chapter 11: Repeated-measures designs (GLM 4)
*Effect sizes:*Field, Chapter xx, section:**Advanced reading****Software****Reporting - Advice****Power:****Effect size:**