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Unformatted text preview: Overview of lecture • • • • • What is ANCOVA? Partitioning Variability Assumptions Examples Limitations Analysis of covariance • Analysis of Covariance is used to achieve statistical control of error when experimental control of error is not possible. • The Ancova adjusts the analysis in two ways:• reducing the estimates of experimental error • 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 • In most experiments the scores on the covariate are collected before the experimental treatment • eg. pretest scores, exam scores, IQ etc • In some experiments the scores on the covariate are 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 since it is assumed that the treatment and covariate are independent. Partitioning variability in ANOVA • In analysis of variance the variability is divided into two components • Experimental effect • Error  experimental and individual differences Error Effect C82MST Statistical Methods 2  Lecture 9 3 C82MST Statistical Methods 2  Lecture 9 4 Partitioning variability in ANCOVA • In ancova we partition variance into three basic components: • Effect • Error • Covariate Estimating treatment effects • When covariate scores are available we have information about differences between treatment groups that existed before the experiment was performed • Ancova uses linear regression to estimate the size of treatment effects given the covariate information • The adjustment for group differences can either increase of decrease depending on the dependent variables relationship with the covariate. Error Effect Covariate C82MST Statistical Methods 2  Lecture 9 5 C82MST Statistical Methods 2  Lecture 9 6 1 Error variability in ANOVA • In between groups analysis of variance the error variability comes from the subject within group deviation from the mean of the group. • It is calculated on the basis of the S/A sum of squares
dependent variable Error variability in ANCOVA
• In regression the residual sum of squares is based on the deviation of the score from the regression line. • The residual sum of squares will be smaller than the S/A sum of squares • This is how ANCOVA works
dependent variable A1 A1 † †
covariate 7
C82MST Statistical Methods 2  Lecture 9 covariate
C82MST Statistical Methods 2  Lecture 9 8 Assumptions of ANCOVA • There are a number of assumptions that underlie the analysis of covariance • All the assumptions that apply to between groups ANOVA • normality of treatment levels • independence of variance estimates • homogeneity of variance • random sampling • Two assumptions specific to ANCOVA • The assumption of linear regression • The assumption of homogeneity of regression coefficients
C82MST Statistical Methods 2  Lecture 9 The assumption of linear regression • This states that the deviations from the regression equation across the different levels of the independent variable have • normal distributions with means of zero • homoscedasticity. • If linear regression is used when the true regression is curvilinear then • the ANCOVA will be of little use. • adjusting the means with respect to the linear equation will be pointless 9 C82MST Statistical Methods 2  Lecture 9 10 Homoscedasticity  Equal Scatter
30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 Covariate Scores (Group A)
Dependent Variable
Dependent Variable The homogeneity of regression coefficients
• Homogeneity of Regression Coefficients • The regression coefficients for each of the groups in the independent variable(s) should be the same. • Glass et al (1972) have argued that this assumption is only important if the regression coefficients are significantly different • We can test this assumption by looking at the interaction between the independent variable and the covariate
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C82MST Statistical Methods 2  Lecture 9 35 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70
Covariate Scores (Group B) Level 1 Level 2 Level 3 Dependent Variable Dependent Variable 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 Covariate Scores (Group A) 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 Covariate Scores (Group B) DV homoscedasticity Covariate heteroscedasticity
C82MST Statistical Methods 2  Lecture 9 12 2 An Example Ancova • A researcher is looking at performance on crossword clues. • Subjects have been grouped into three vocabulary levels. • An anova & tukeys on this data finds that the high group and low groups are different An Example Ancova • However amount of experience solving crosswords might make a difference. • Plotting the scores against the age we obtain this graph. • Ancova produces a significant effect of age and vocabulary. This time all the groups are significantly different C82MST Statistical Methods 2  Lecture 9 13 C82MST Statistical Methods 2  Lecture 9 14 Example Results  ANOVA
Mean 5.3750 6.7500 7.6250 6.5833 Std. Error .32390 .52610 .32390 .29437 Example Results  ANCOVA low medium high Total low medium high Mean Std. Error 5.679 .153 6.699 .151 7.372 .152 Between Groups Within Groups Total Sum of Squares 20.583 27.250 47.833 df 2 21 23 Mean Square 10.292 1.298 F 7.931 Sig. .003 Source AGE GROUP Error Total
15 Sum of Squares 23.623 11.042 3.627 1088.000 df 1 2 20 24 Mean F Square 23.623 130.268 5.521 .181 30.445 Sig. .000 .000 C82MST Statistical Methods 2  Lecture 9 C82MST Statistical Methods 2  Lecture 9 16 Example Results  Post hoc Tukey tests A Teaching Intervention Example • Two groups of children either use maths training software or they do not. • After using (or not using) the software the participants maths abilities are measured using a standardised maths test low low medium medium high high Mean Difference 1.3750 2.2500 .8750 Mean Difference 1.021 1.694 .673 Sig. .062 .002 .295 Sig. .000 .000 .005 low low medium medium high high C82MST Statistical Methods 2  Lecture 9 17 C82MST Statistical Methods 2  Lecture 9 18 3 Posttest results  Using ttest Problems with the design? • It is quite possible that prior mathematical ability varies between the two groups of children • This needs to be taken into account • Prior to using the software the participants’ maths abilities are measured using a standardised maths test GROUP software no software Mean 14.0000 14.1500 Std. Deviation 4.18015 5.54669 t .097 df Sig. (2tailed) 38 .924 Mean Difference .1500 C82MST Statistical Methods 2  Lecture 9 19 C82MST Statistical Methods 2  Lecture 9 20 Posttest results  Using ANCOVA
Mean software 16.236 no software 11.914 Source PRETEST GROUP Error SS 703.676 145.472 212.874 df 1 1 37 40 MS 703.676 145.472 5.753 Std. Error .573 .573 F 122.307 25.285 Sig. .000 .000 Limitations to analysis of covariance • As a general rule a very small number of covariates is best • Correlated with the dv • Not correlated with each other (multicollinearity) • Covariates must be independent of treatment • Data on covariates be gathered before treatment is administered • Failure to do this often means that some portion of the effect of the IV is removed from the DV when the covariate adjustment is calculated. Total 8841.000 • NB the means have been adjusted for the pretest 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 ...
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This note was uploaded on 01/30/2011 for the course MAT 219 taught by Professor Ab during the Spring '10 term at Coast Guard Academy.
 Spring '10
 AB
 Covariance, Variance

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