Lecture 9: One Way ANOVA Between Subjects Laura McAvinue School of Psychology Trinity College Dublin

Analysis of Variance A statistical technique for testing for differences between the means of several groups One of the most widely used statistical tests T-Test Compare the means of two groups Independent samples Paired samples ANOVA No restriction on the number of groups

Is the mean of one group significantly different to the T-test Group 1   Group 2   Mean Mean Is the mean of one group significantly different to the mean of the other group? t-test: H0 - 1= 2 H1: 1 2

F-test Group 2   Group 3   Group 1   Mean Mean Mean Is the mean of one group significantly different to the means of the other groups?

Analysis of Variance One way ANOVA Factorial ANOVA More than One Independent Variable One Independent Variable Between subjects Repeated measures / Within subjects Two way Three way Four way Different participants Same participants

A few examples… Between subjects one way ANOVA The effect of one independent variable with three or more levels on a dependent variable What are the independent & dependent variables in each of the following studies? The effect of three drugs on reaction time The effect of five styles of teaching on exam results The effect of age (old, middle, young) on recall The effect of gender (male, female) on hostility

Rationale Let’s say you have three groups and you want to see if they are significantly different… Recall inferential statistics Sample Population Your question: Are these 3 groups representative of the same population or of different populations?

measure effect of manipulation on a DV Population Draw 3 samples 1 2 Did the manipulation alter the samples to such an extent that they now represent different populations? 3 Drug 1 Drug 2 Drug 3 Manipulate the samples DV µ1 µ2 µ3 measure effect of manipulation on a DV

Recall sampling error & the sampling distribution of the mean… The means of samples drawn from the same population will differ a little due to random sampling error When comparing the means of a number of groups, your task … Difference due to a true difference between the samples (representative of different populations)? Difference due to random sampling error (representative of the same population)? If a true difference exists, this is due to your manipulation, the independent variable

Steps of NHST Specify the alternative / research hypothesis At least one mean is significantly different from the others At least one group is representative of a separate population Set up the null hypothesis The hypothesis that all population means are equal All groups are representative of the same population Omnibus Ho: µ1= µ2 = µ3

Steps of NHST F statistic Collect your data Run the appropriate statistical test Between subjects one way ANOVA Obtain the test statistic & associated p-value F statistic Compare the F statistic you obtained with the distribution of F when Ho is true Determine the probability of obtaining such an F value when Ho is true

Steps of NHST Decide whether to reject or fail to reject Ho on the basis of the p value If the p value is very small (<.5), reject Ho… Conclude that at least one sample mean is significantly different to the other means… Not all groups are representative of the same population

How is ANOVA done? Assume Ho is true Assume that all three groups are representative of the same population Make two estimates of the variance of this population If Ho is true, then these two estimates should be about the same If Ho is false, these two estimates should be different

Two estimates of population variance Within group variance Pooled variability among participants in each treatment group Between group variance Variability among group means If Ho is true… Between Groups Variance Within Groups Variance = 1 If Ho is false… Between Groups Variance Within Groups Variance > 1

Calculations Step… 1: Sum of squares 2: Degrees of freedom 3: Mean square 4: F ratio 5: p value

Total Variance In data SStotal Within groups Variance SSwithin Between groups variance SSbetween

SStotal ∑ (xij - Grand Mean )2 Based on the difference between each score and the grand mean The sum of squared deviations of all observations, regardless of group membership, from the grand mean

SSbetween n∑ (Group meanj - Grand Mean )2 Based on the differences between groups Related to the variance of the group means The sum of squared deviations of the group means from the grand mean, multiplied by the number of observations in each group

SSwithin ∑ (xij - Group Meanj )2 Based on the variability within each group Calculate SS within each group & add The sum of squared deviations within each group … or … SStotal - SSbetween

Degrees of Freedom Total variance Between groups variance Total no. of observations - 1 Between groups variance K – 1 No. of groups – 1 Within groups variance k (n – 1) No. of groups (no. in each sample – 1) What’s left over!

Mean Square SS / df The average variance between or within groups An estimate of the population variance MSbetween SSgroup / dfgroup MSwithin SSwithin / dfwithin

F Ratio MSbetween MSwithin If Ho is true, F = 1 If Ho is false, F > 1

F F 1 > 1 F MSbetween MSwithin Treatment effect + Differences due to chance Differences due to chance F If treatment has no effect… 0 + Differences due to chance Differences due to chance F 1 If treatment has effect… EFFECT > 0 + Differences due to chance Differences due to chance > 1 F

MSBG MSBG MSBG MSWG MSWG MSWG Variance within groups> variance between groups F<1 Fail to reject Ho If there is more variance within the groups, then any difference observed is due to chance Variance within groups= Variance between groups F =1 Fail to reject Ho If both sources of variance are the same, then any difference observed is due to chance Variance within groups < variance between groups F >1 Reject Ho The more the group means differ relative to each other the more likely it is that the differences are not due to chance.

Size of F How much greater than 1 does F have to be to reject Ho? Compare the obtained F statistic to the distribution of F when Ho is true Calculate the probability of obtaining this F value when Ho is true p value If p < .05, reject Ho Conclude that at least one of your groups is significantly different from the others

ANOVA table Source of variation SS df MS F p Between groups n∑ (Group meanj - Grand Mean )2 K - 1 SSBG / dfBG MSBetween MSWithin Prob. of observing F-value when Ho is true Within groups ∑ (xij - Group Meanj )2 K(n – 1) SSWG / dfWG Total ∑ (xij - Grand Mean )2 N - 1

A few assumptions… Data in each group should be… Interval scale Normally distributed Histograms, box plots Homogeneity of variance Variance of groups should be roughly equal Independence of observations Each person should be in only one group Participants should be randomly assigned to groups

Multiple Comparison Procedures Obtain a significant F statistic Reject Ho & conclude that at least one sample mean is significantly different from the others But which one? H1: µ1 ≠ µ2 ≠ µ3 H2: µ1 = µ2 ≠ µ3 H3: µ1 ≠ µ2 = µ3 Necessary to run a series of multiple comparisons to compare groups and see where the significant differences lie

Problem with Multiple Comparisons Making multiple comparisons leads to a higher probability of making a Type I error The more comparisons you make, the higher the probability of making a Type I error Familywise error rate The probability that a family of comparisons contains at least one Type I error

Problem with Multiple Comparisons familywise = 1 - (1 - )c c = number of comparisons Four comparisons run at  = .05 familywise = 1 - (1 - .05)4 = 1 - .8145 = .19 You think you are working at  = .05, but you’re actually working at  = .19

Post hoc tests Bonferroni Procedure Four comparisons at  = .05  / c Divide your significance level by the number of comparisons you plan on making and use this more conservative value as your level of significance Four comparisons at  = .05 .05 / 4 = .0125 Reject Ho if p < .0125

Post hoc tests Note: Restrict the number of comparisons to the ones you are most interested in Tukey Compares each mean with each other mean in a way that keeps the maximum familywise error rate to .05 Computes a single value that represents the minimum difference between group means that is necessary for significance

Effect Size A statistically significant difference might not mean anything in the real world Eta squared Percentage of variability among observations that can be attributed to the differences between the groups

When comparing two groups Meantreat – Meancontrol SDcontrol A little less biased… Omega squared How big is big? Similar to correlation coefficient Cohen’s d When comparing two groups Meantreat – Meancontrol SDcontrol