Overview of lecture Analysis of covariance • What is ANCOVA? • Analysis of Covariance is used to achieve statistical • Partitioning Variability control of error when experimental control of error is not possible. • Assumptions • The Ancova adjusts the analysis in two ways:- • Examples • reducing the estimates of experimental error • Limitations • adjusting treatment effects with respect to the covariate C82MST Statistical Methods 2 - Lecture 9 1 C82MST Statistical Methods 2 - Lecture 9 2 Analysis of covariance Partitioning variability in ANOVA • In most experiments the scores on the covariate are • In analysis of variance the variability is divided into two collected before the experimental treatment components • eg. pretest scores, exam scores, IQ etc • Experimental effect • In some experiments the scores on the covariate are • Error - experimental and individual differences collected after the experimental treatment • e.g.anxiety, motivation, depression etc. • It is important to be able to justify the decision to collect the covariate after the experimental treatment Error Effect since it is assumed that the treatment and covariate are independent. C82MST Statistical Methods 2 - Lecture 9 3 C82MST Statistical Methods 2 - Lecture 9 4 Partitioning variability in ANCOVA Estimating treatment effects • In ancova we partition variance into three basic • When covariate scores are available we have components: information about differences between treatment • Effect groups that existed before the experiment was performed • Error • Ancova uses linear regression to estimate the size of • Covariate treatment effects given the covariate information • The adjustment for group differences can either Error increase of decrease depending on the dependent Effect Covariate variables relationship with the covariate. C82MST Statistical Methods 2 - Lecture 9 5 C82MST Statistical Methods 2 - Lecture 9 6 1 Error variability in ANOVA Error variability in ANCOVA • In between groups analysis of variance the error • In regression the residual sum of squares is based on the variability comes from the subject within group deviation of the score from the regression line. • The residual sum of squares will be smaller than the S/A sum deviation from the mean of the group. of squares • It is calculated on the basis of the S/A sum of squares • This is how ANCOVA works variable variable A1 A1 dependent dependent † † covariate covariate C82MST Statistical Methods 2 - Lecture 9 7 C82MST Statistical Methods 2 - Lecture 9 8 Assumptions of ANCOVA The assumption of linear regression • There are a number of assumptions that underlie the • This states that the deviations from the regression analysis of covariance equation across the different levels of the independent • All the assumptions that apply to between groups variable have ANOVA • normal distributions with means of zero • normality of treatment levels • homoscedasticity. • independence of variance estimates • If linear regression is used when the true regression is • homogeneity of variance curvilinear then • random sampling • the ANCOVA will be of little use. • Two assumptions specific to ANCOVA • adjusting the means with respect to the linear • The assumption of linear regression equation will be pointless • The assumption of homogeneity of regression coefficients C82MST Statistical Methods 2 - Lecture 9 9 C82MST Statistical Methods 2 - Lecture 9 10 Homoscedasticity - Equal Scatter The homogeneity of regression coefficients 35 35 30 30 iable 25 25 • Homogeneity of Regression Var 20 20 Coefficients 15 15 • The regression coefficients Level 1 10 10 Level 2 5 5 for each of the groups in Level 3 Dependent 0 Dependent Variable 0 the independent 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 variable(s) should be the Covariate Scores (Group A) Covariate Scores (Group B) same. homoscedasticity • Glass et al (1972) have argued that this DV 35 35 assumption is only 30 30 important if the regression 25 25 20 20 coefficients are 15 15 significantly different 10 10 5 5 • We can test this 0 0 Dependent Variable Dependent Variable 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 assumption by looking at Covariate Scores (Group A) Covariate Scores (Group B) the interaction between the Covariate independent variable and heteroscedasticity the covariate C82MST Statistical Methods 2 - Lecture 9 11 C82MST Statistical Methods 2 - Lecture 9 12 2 An Example Ancova An Example Ancova • A researcher is looking • However amount of at performance on experience solving crossword clues. crosswords might make a difference. • Subjects have been • Plotting the scores against grouped into three the age we obtain this vocabulary levels. graph. • An anova & tukeys on • Ancova produces a this data finds that the significant effect of age and high group and low vocabulary. This time all groups are different the groups are significantly different C82MST Statistical Methods 2 - Lecture 9 13 C82MST Statistical Methods 2 - Lecture 9 14 Example Results - ANOVA Example Results - ANCOVA Mean Std. Error Mean Std. Error low 5.3750 .32390 low 5.679 .153 medium 6.7500 .52610 medium 6.699 .151 high 7.6250 .32390 high 7.372 .152 Total 6.5833 .29437 Source Sum of df Mean F Sig. Sum of df Mean F Sig. Squares Square Squares Square AGE 23.623 1 23.623 130.268 .000 Between Groups 20.583 2 10.292 7.931 .003 GROUP 11.042 2 5.521 30.445 .000 Within Groups 27.250 21 1.298 Error 3.627 20 .181 Total 47.833 23 Total 1088.000 24 C82MST Statistical Methods 2 - Lecture 9 15 C82MST Statistical Methods 2 - Lecture 9 16 Example Results - Post hoc Tukey tests A Teaching Intervention Example Mean Sig. • Two groups of children either use maths training Difference software or they do not. low medium -1.3750 .062 • After using (or not using) the software the participants low high -2.2500 .002 maths abilities are measured using a standardised medium high -.8750 .295 maths test Mean Sig. Difference low medium -1.021 .000 low high -1.694 .000 medium high -.673 .005 C82MST Statistical Methods 2 - Lecture 9 17 C82MST Statistical Methods 2 - Lecture 9 18 3 Post-test results - Using t-test Problems with the design? • It is quite possible that prior mathematical ability varies GROUP Mean Std. Deviation between the two groups of children software 14.0000 4.18015 • This needs to be taken into account no software 14.1500 5.54669 • Prior to using the software the participants’ maths abilities are measured using a standardised maths test t df Sig. (2-tailed) Mean Difference -.097 38 .924 -.1500 C82MST Statistical Methods 2 - Lecture 9 19 C82MST Statistical Methods 2 - Lecture 9 20 Post-test results - Using ANCOVA Limitations to analysis of covariance Mean Std. Error • As a general rule a very small number of covariates is software 16.236 .573 best no software 11.914 .573 • Correlated with the dv • Not correlated with each other (multi-collinearity) Source SS df MS F Sig. • Covariates must be independent of treatment PRETEST 703.676 1 703.676 122.307 .000 • Data on covariates be gathered before treatment is GROUP 145.472 1 145.472 25.285 .000 administered Error 212.874 37 5.753 • Failure to do this often means that some portion of Total 8841.000 40 the effect of the IV is removed from the DV when the covariate adjustment is calculated. • NB the means have been adjusted for the pre-test covariate C82MST Statistical Methods 2 - Lecture 9 21 C82MST Statistical Methods 2 - Lecture 9 22 Next Week • MANOVA - Multivariate Analysis of Variance C82MST Statistical Methods 2 - Lecture 9 23 4.