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One Way Analysis of Variance:
Group Variance and Treatment Effects
The Sampling Distribution of F/Rationale of the one way ANOVA
Interpreting A Significant F
Interpreting A Significant F Continued....
100
What is a one way ANOVA? Why is it called one way?
- It is an extension of the two sample t-test - Useful in assessing the significance of differences between 2 or ore group means - The 2 or more samples being compared differ on a single IV
100
What are the 2 sources of variability? What is each one caused by?
- Within group variance - Caused by: - Individual differences - Measurement error - Between group Variance - Caused by: - Individual differences - Measurement error - Effects of the IV on the DV
100
What is the sampling distribution of F? What does it look like? (Draw it)
It indicates the likelihood of seeing any particular value of F in a set of identically treated samples
100
What tests do you use to determine which specific means differ?
Post Hoc Comparison Priori Comparison
100
How do post hoc comparisons set the probability of Type 1 errors?
At chosen level for entire set of post hoc comparisons
200
One way ANOVAS require the DV to be measured at what levels? What do one way ANOVAS assume?
- Interval - Ratio - DV is normally distributed - Groups show approx. equal variance
200
What happens to the 2 types of variance if the IV affects the DV? What happens if the IV doesn’t have an effect on the DV?
- Between group variance > than within group variance (There is a treatment effect) - Between group and within group variances should be equal (B/c determined by the same 2 factors)
200
When all samples are identical, F=? What is the most frequently occurring value of F? Are high values of F likely?
F = 0 F = 1 Unlikely, because high values occur when sample means differ substantially (unlikely when all samples receive identical treatments)
200
What are the two types of Post Hoc Comparisons? Are they powerful?
Tukey’s HSD Scheffe Test Not really
200
What are the restrictions for a priori (planned) comparison? How does the priori comparison compare to the post hoc comparisons?
Restrictions: - Can only be used in testing differences that were predicted before data was collected - Only independent comparisons may be made More powerful
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What are fixed effects models and random effects models?
- Fixed Effects Model: IV may have been selected as the only levels that are of interest - Random Effects Model: IV may have been randomly sampled to rep. a broader assortment of levels
300
What does the F statistic reflect? How is it computed?
- The amount of separation between the groups frequency distributions - Computed as ratio of between group variance to within group variance (F = MSbetween / MSwithin)
300
What does the shape of the sampling distribution of F depend on?
Sample sizes Number of samples
300
When is Tukey’s HSD computed? How do you know if there is a statistical significance?
Used in making all possible pairwise comparisons between group means If the difference obtained > HSD
300
How do you compute a priori comparison?
Compute C Compute tobtained Compare tobtained to tcritical (Appendix B, df = N-K) Statistically significant if tobtained > tcritical
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What are completely randomized ANOVA designs and repeated measures/randomized block ANOVA designs?
- Any given case may be exposed to only 1 level of the IV - Each case may be exposed to all levels of the IV
400
What does the sum of squares represent? How is it calculated? What is it influenced by? What do mean squares represent? How is it calculated? What is it influenced by?
- Variance - SSwithin = subtract a groups mean from each score in the group, square the difference, sum the squares and sum these values - SSbetween = subtract grand mean from each group mean, square the differences, multiply each by size of the group, sum these values - SStotal = subtract grand mean from each score, square the differences, then sum the squares - Influenced by: data variability, sample size and number of samples - MSwithin = SSwithin/dfwithin - MSbetween = SSbetween/dfbetween - No longer influenced by sample size and number of samples
400
What are the 3 steps of the One way ANOVA? Expand on Steps 1 and 2.
- Step 1: Null and Alternative hypotheses - H0 = Differences observed due to sampling error - H1 = Differences observed too big, groups probably received different treatments to produce differences - Step 2: Test Statistic - F statistic - Value of F increase as treatment effects increase - Indicates: how much the obtained between group differences deviate from what would be expected - Step 3: Determine the Probability of the Test Statistic - Appendix G: Critical Values of F
400
When is the Scheffe test computed? How does it compare in power to Tukey’s HSD?
Can be used in comparing means as well as other combinations (more generally useful) Less Powerful
400
What does Omega square tell you?
The treatment effect strength
500
What are the advantages and disadvantages of a one way ANOVA vs multiple pairwise comparisons?
- Advantages: Computational ease, reduced error rates - Disadvantage: It doesn’t indicate which specific means differ significantly
500
What is the ANOVA sampling table? What does it look like?
Standardized format of results
500
What does the following indicate? Fobtained > Fcritical Fobtained < Fcritical
Reject H0 Fobtained would be statistically significant Accept H0 Fobtained not statistically significant
500
When computing C for the Scheffe test, weights must take on different values. What must the sum of the weights be? What are means weighted as if they are not included in the comparison? What happens to means that are grouped together? (In terms of their weight and magnitude?)
Sum = 0 Means not included w = 0 Grouped means get same weights of same size and magnitude
500
Omega square can vary in value from _____ to ____. What do these values tell you?
0: no treatment effect 1: all variability in scores attributed to different levels of IV examined