Power of a Test/Comparing Means with ANOVA (4/6)

Type 1 error: Rejecting Ho, when you shouldn’t have (i.e., a false positive); and the probability is α\alpha

Type 2 error: Failing to Reject Ho; the probability is β\beta

The Power of a Test: 1 - β\beta

Keep α\alpha and β\beta low, but there are inherent tradeoffs

Ex: β\beta =.15

p(type 2 error) = β\beta = .15

This is the probability of correctly rejecting the false null hypothesis is .85

p(type1error

  • Rejecting the null means the test has found a significant difference

TO increase the power

  • Increase sample size

  • Increase α\alpha - but Type 1 error is more likely

  • Decrease Standard deviation

Comparing Means with ANOVA (4/6)

  • To compare means from 2 or more groups; uses variance in the comparison

Variance: The spread of the data

Variability between samples due to different treatment

ex: different medicine, different diet, textbook

Variability within samples due to regular sampling error simply based on difference between subjects that ended up in the sample

F distribution, right skewed

Always use α\alpha = .05

Repeated Measures: Use the same group of subjects with each treatment

ex: Instead of 4 different groups of patients testing 4 drugs, let 1 group of patients test all 4 different drugs and compare the results

Setup for ALL ANOVA tests:

Ho: μ1=μ2=μ3=μ4\mu1=\mu2=\mu3=\mu4

Ha: Not all means are equal OR at least one mean is different

Conditions:

  1. Random Sample, <10% of population

  2. Observations are independent

  3. Observations should be nearly normal

  4. Variances should be about equal

If yes (reject null), there is a difference. Find the difference doing post hoc(after the fact), pairwise t-tests

Homoscedasticity must be assumed for the ANOVA to be valid, ensuring that the error variances are equal across the groups being compared.

Heteroscedasticity indicates that the variances are not equal across groups, which can lead to incorrect conclusions if ANOVA is applied without addressing this issue. When heteroscedasticity is present, it may be necessary to use alternative statistical methods or to apply transformations to the data to stabilize the variances before conducting an ANOVA.