stats week 3

stats week 3

1. one way ANOVA

   • used when we have 1 IV with more than 2 levels - estimates whether the population means under the different levels of the IV are different
     • essentially just an extension of the t-test
   • why? - when P < .05, we reject null - but theres still 5% chance we've made type 1 error - with 3 IV levels, we'd run 3 t-tests and for each there would be a 5% chance of error, so overall chance would be higher
     • family wise error rate = the probability that at least one of a family of comparisons, run on the same data, will result in a type 1 error - it provides a corrected significant level, expressing the probability of making a type 1 error = a" = 1 (1-a)^c = a" = error rate for family of comparisons, (1-a) = level of confidence we're working to for each comparison (so a is .05) c = number of comparisons
       • e.g for 3 IV levels - 1 - (1 - 0.05)^3 = 0.143 - i.e 14% chance of type 1 error
         • ANOVA controls the family wise error rate
   • assumptions - normality, homogeneity of variance (Lavenes stat tests for this, welch's F corrects for this) - sample size equivalent, independence of observations
     • equality of variance - null hypothesis = no difference between the variance under each level of the IV - if P is less than .05 we reject the null (i.e if lavenes stat is significant)
       • contributions to variance same for independent t-test
         • total variance = variance between iv levels (includes variance explained by our model + variance within IV levels (residual or error variance)
           • partitioning variance - calculate means for each IV level, calculate grand mean (sum of IV level means divided by no. of IV levels) - calculate the within IV levels variance (sum of squared differences between individual values and the corresponding IV level mean) - calculate the between levels variance (sum of squared differences between each IV level mean and the grand mean
             • SSm = model sum of squares ( sum of squared differences between IV level means and the grand mean) SSr = residual sum of squares - sum of squared differences between individual values and corresponding IV level mean - Sum + SSr = SSt (Set = total sum of squares)
               • SSm / DFm = MSm ( mean square between level variance)
                 • SSr / DFr = MSr (mean square within level variance)
                   • F = MSm / MS r
                     • df = no. measurements / no. of parameters estimates - N is total sample size, k is number of IV levels - so df for model variance (between) = k - 1, whereas degree of freedom for residual variance (within) is N - k

2. post hoc tests

   • used to assess which IV level means differ - should only be used when F is significant - run as t-test but include correction for multiple comparisons
     • Boferroni = very low type 1 error risk, very high type 2 error risk (very conservative)
       • Least significant difference - type 1 error risk high, type 2 low (liberal)
         • Turkey Honestly significant difference - type1 error risk low, type 2 high, (reasonably conservative)

3. effect size

   • for AONVA we calculate 2 different effect sizes - partial n2 - how much variance in the DV is explained by the manipulation of the IV overall - coned d - the magnitude of the difference between pairs of IV level mean, expressed in SD units
     • Cohen's d - small >0.2, medium = >0.5, large = >0.8
       • partial n2 - small = >0.01, medium = > 0.06, Large = >0.14
         • partial n2 - what proportion of the variance in the DV is accounted for by the manipulation of the IV? - SSm / SSm + SSr e.g answer of 0.204 = 20% of variance in verbal short term memory performance was accounted for by the mnemonic strategy used
       • the magnitude of the difference between iV level means, expressed in SD units - needs to be calculated for each pair of IV level means - e.g the means differ by _ SDs, demonstrating a __ effect size