Forfatter av avsnitt: Danielle J. Navarro and David R. Foxcroft

Kovariansanalyse (ANCOVA)

A variation in ANOVA is when you have an additional continuous variable that you think might be related to the dependent variable. This additional variable can be added to the analysis as a covariate, in the aptly named analysis of covariance (ANCOVA). In ANCOVA the values of the dependent variable are “adjusted” for the influence of the covariate, and then the “adjusted” score means are tested between groups in the usual way. This technique can increase the precision of an experiment, and therefore provide a more “powerful” test of the equality of group means in the dependent variable. Although the covariate itself is typically not of any experimental interest, adjusting for the covariate can reduce the error variance, and thereby increase precision. This means that a failure to inappropriately reject the null hypothesis (Type II error) becomes less likely.

Despite this advantage, ANCOVA runs the risk of undoing real differences between groups , and this should be avoided. Look at Fig. 176, for example, which shows a plot of Statistics anxiety against age and shows two distinct groups – students who have either an Arts or Science background. ANCOVA with age as a covariate might lead to the conclusion that statistics anxiety does not differ in the two groups. Would this conclusion be reasonable – probably not because the ages of the two groups do not overlap and analysis of variance has essentially “extrapolated into a region with no data” (Everitt, 1996).

Fig. 176 Plott av statistikk for angst mot alder for to forskjellige grupper

Det er klart at man må tenke nøye gjennom en kovariansanalyse med distinkte grupper. Dette gjelder både enveis- og faktorielle design, ettersom ANCOVA kan brukes med begge.

Kjører ANCOVA i jamovi

A health psychologist was interested in the effect of routine cycling and stress on happiness levels, with age as a covariate. Open the ancova data set in jamovi and then, to undertake an ANCOVA, select Analyses → ANOVA → ANCOVA to open the ANCOVA analysis window (Fig. 177). Highlight the dependent variable happiness and transfer it into the Dependent Variable text box. Highlight the independent variables stress and commute and transfer them into the Fixed Factors text box. Highlight the covariate age and transfer it into the Covariates text box. Then, click on Estimated Marginal Means to bring up the plots and tables options.

Fig. 177 Analysepanel for ANCOVA i jamovi som viser variabelboksene for å tilordne Dependent Variable, Fixed Factors og Covariates

An ANCOVA table showing Tests of Between-Subjects Effects is produced in the jamovi results panel (Fig. 178). The F-value for the covariate age is significant at p = 0.023, suggesting that age is an important predictor of the dependent variable, happiness. When we look at the estimated marginal mean scores (Fig. 179), adjustments have been made (compared to an analysis without the covariate) because of the inclusion of the covariate age in this ANCOVA. A plot (Fig. 180) is a good way of visualising and interpreting the significant effects.

Fig. 178 Utgave fra ANCOVA for lykke som en funksjon av stress og pendlingsmetode, med alder som kovariat i jamovi

Fig. 179 Tabell med de estimerte randgjennomsnittene i ANCOVA: Her vises gjennomsnittlig lykkenivå som en funksjon av stress og pendlingsmetode (justert for kovariaten alder) med 95%-konfidensintervall

The F-value for the main effect stress (52.61) has an associated probability of p < 0.001. The F-value for the main effect commute (42.33) has an associated probability of p < 0.001. Since both of these are less than the probability that is typically used to decide if a statistical result is significant (p < 0.05) we can conclude that there was a significant main effect of stress: F(1,15) = 52.61, p < 0.001, and a significant main effect of commuting method: F(1,15) = 42.33, p < 0.001. A significant interaction between stress and commuting method was also found: F(1,15) = 14.15, p = 0.002.

In Fig. 180 we can see the adjusted, marginal, mean happiness scores when age is a covariate in an ANCOVA. In this analysis there is a significant interaction effect, whereby people with low stress who cycle to work are happier than people with low stress who drive and people with high stress regardless of whether they cycle or drive to work. There is also a significant main effect of stress – people with low stress are happier than those with high stress. And there is also a significant main effect of commuting behaviour – people who cycle are happier, on average, than those who drive to work.

Fig. 180 Plott med de estimerte randgjennomsnittene i ANCOVA: Her vises gjennomsnittlig lykkenivå som en funksjon av stress og pendlingsmetode

En ting du må være oppmerksom på, er at hvis du vurderer å inkludere en kovariat i ANOVA-en din, er det en ekstra forutsetning: Forholdet mellom kovariaten og den avhengige variabelen skal være likt for alle nivåer av den uavhengige variabelen. Dette kan kontrolleres ved å legge til en interaksjonsterm mellom kovariaten og hver av de uavhengige variablene i jamovi Model → Model terms. Hvis interaksjonseffekten ikke er signifikant, kan den fjernes. Hvis den er signifikant, kan en annen og mer avansert statistisk teknikk være hensiktsmessig (noe som ligger utenfor denne bokens omfang, så det kan være lurt å rådføre seg med en vennligsinnet statistiker).