Stats Test 3 (Ch. 12- 14) Lab Notes
MSB Between- Group Variance (One- Way Between- Groups ANOVA)
MSB = SSB/ dfB SSB= (number of scores in each group) * (mean of that group - grand mean) squared dfB= k-1
MSB Between- Group Variance (One- Way Within- Subjects ANOVA)
MSB = SSB/ dfB SSB= (number of scores in each group) * (mean of that group - grand mean) squared dfB= k-1
MSW Within- Group Variance (One- Way Within- Subjects ANOVA)
MSE = SSerror / df denominator -First calculate SSW= (particular score in group - that groups mean) squared then added for all scores -Then calculate SSsubjects (between persons) SSsub= (number of groups) * (participant's mean- GM) squared -Participant's mean: Sum across the row of participants and divide by k, the number of groups -Then calculate SSerror= SSwithin - SSsubjects -Finally calculate df denominator = (N-1)(k-1)
MSW Within- Group Variance (One- Way Between- Groups ANOVA)
MSW = SSW/ dfW SSW= (particular score in group - that groups mean) squared then added for all scores dfW= N-k *This is the long one
If MSB is Larger Than MSW
Larger F and therefore more likely to find significant difference between groups
Main Effect
-A significant IV is called a main effect -This is basically just saying that at least one of your groups in your IV is significantly different from the rest of them on your DV (what you are measuring)
Interaction
-If IV1 and IV2 both influence your DV, this is called an interaction -This is basically saying that being a member of certain groups in IV1 AND IV2 results significant differences on your DV (what you are measuring) -COMBINED EFFECT OF YOUR IVS ON YOUR DVS
Reject or Fail to Reject ANOVA
-If there is no difference in the means, the between-group variance will be equal to the within-group variance THEREFORE fail to reject null hypothesis -When the means differ significantly, the between-group variance will be larger than the within-group variance THEREFORE reject null hypothesis
Alpha Inflation
-If we set alpha-level at .05 for every independent samples t-test, and we do multiple t-tests ON THE SAME DATA, then we are actually INFLATING our alpha-level -This means that our probability of committing a Type I error is actually much larger than the what we are setting our alpha to be, in this case .05 -Example: If you performed 6 t-tests with alpha = .05, your probability of committing a Type I error would actually be 25% - not 5%
F Test
-In the F test, two different estimates of the population variance are made -Between-group variance: finding the variance of the means -Within-group variance: computing the variance using all the data and is not affected by differences in the means -F = between-group var / within-group var
Independent vs. Dependent Variables
-Independent variable (IV): what you are manipulating *For One-way ANOVA, you will only have 1 IV *For factorial ANOVA, you will have 2 or more IVs -Dependent variable (DV): what you are measuring
Post Hoc Tests Between vs. Within
-Previously, for Between-Subjects ANOVA, perform Tukey post-hoc test -Now, for Within-Subjects ANOVA, use the Bonferroni procedure -The Bonferroni procedure adjusts the alpha level depending on how many comparisons you are making... New alpha = Old alpha / number of comparisons
Between- Subjects vs. Within- Subjects
-Previously, with between-groups, our F ratio = MS between / MS within -Accounting for the variance between groups and within each group -Now, with within-groups, our F ratio accounts for the between and within group variance, but it also accounts for the between persons variance
Decision
-Reject the null hypothesis if test value is greater than or equal to critical value -Fail to reject null hypothesis if test value is less than critical value
Decision Using SPSS
-When we use SPSS, we do not look up a critical F in the F table and compare it to our observed (calculated) F -Instead, we get a p-value, and we compare this p-value to our alpha... If p > .05 (or whatever alpha is defined to be), fail to reject null If p ≤.05 (or whatever alpha is defined to be), reject null
Find F Critical Value (One- Way Within- Subjects ANOVA)
-k = # of groups we are comparing (k = 3) -N = # of total participants -Degrees of freedom between- groups (numerator) dfBG = k-1 -Degrees of freedom error (denominator) dfE= (N-1) (k-1) -You will need dfBG, dfE, and alpha to find the F critical value
Find F Critical Value (One- Way Between- Groups ANOVA)
-k = # of groups we are comparing (k = 3) -N = # of total participants -Degrees of freedom between- groups (numerator) dfBG = k-1 -Degrees of freedom within- groups (denominator) dfWG= N-k -You will need dfBG, dfWG, and alpha to find the F critical value
Find F Test Value (One- Way Between- Groups ANOVA)
1. Find the grand mean (GM) 2. Calculate SSB and dfnum to find MSB (Between-group variance) 3. Calculate SSw and dfden to find MSw (Within-group variance) 4. Use MSw and MSB to calculate your test statistic, F
Find F Test Value (One- Way Within- Subjects ANOVA)
1. Find the grand mean (GM) 2. Calculate SSB and dfnum to find MSB (Between-group variance) 3. Calculate SSw, SSsubjects, and dfden to find MSE (Within-group variance) 4. Use MSw and MSB to calculate your test statistic, F
Steps of One- Way Between- Groups ANOVA
1. State hypotheses 2. Find F critical value 3. Find F test value 4. Decide whether to reject or fail to reject 5. Interpret
Steps of One- Way Within- Subjects ANOVA
1. State hypotheses 2. Find F critical value 3. Find F test value 4. Decide whether to reject or fail to reject 5. Interpret
What is Effect Size Telling Us?
Effect size informs us of practical significance, meaning how large or small our effect really is. Statistical significance (the t and F tests) are influenced by amount of participants in a study (N), but effect size is not; therefore, it is just measuring the effect
Effect Size
Eta squared (η2) η2 = SSB / (SSB + SSW)
Calculate F (One- Way Within- Subjects ANOVA)
F = MSBG / MSE
Calculate F (One- Way Between- Groups ANOVA)
F= MSB / MSW
Grand Mean (One- Way Between- Groups ANOVA)
Find the sum of all X's and divide by N (the total number of PARTICIPANTS in the study)
ANOVA Hypotheses (One- Way Between- Groups ANOVA)
H0: μ1 = μ2 = μ3 HA: The means are not all equal, at least one mean differs from the others
ANOVA Hypotheses (One- Way Within- Subjects ANOVA)
H0: μ1 = μ2 = μ3 HA: The means are not all equal, at least one mean differs from the others
What is Sum of Squares Between Measuring?
SSbetween is measuring the variability BETWEEN group means
What is Sum of Squares Within Measuring?
SSwithin is measuring variability around the group mean for each group mean
If MSW is Larger Than MSB
Smaller F and therefore less likely to find significant difference between groups
What is our F-Test Telling us About the Groups in our Data?
The F-test shows whether or not there is a significant difference between at least two means (or groups). A higher MSB and lower MSW gives a larger F-value and vice-versa
Grand Mean (One- Way Within- Subjects ANOVA)
There are two ways you can calculate the grand mean... 1. Add all of the x values and divide by the total number of x values 2. First, calculate a mean for each participant. Then, take the mean of these participant means (add up all of the means and divide by n, or number of participants)
WHY Would You Use ANOVA Instead of a t-Test to Analyze Data?
We would perform an ANOVA over a t-test when we have more than two groups. t-tests are used when we have two groups. ANOVA is used when we have more than two groups. When you have more than two groups, you would NOT ever perform multiple t-tests because your alpha (probability for committing a Type I error, which is finding significance when it does not exist) will be inflated
Basics of ANOVA
With ANOVA, you are estimating the population's variance -When you take samples of the population (which we do in experiments), you are estimating the population variance by comparing the variance among your sample means -If the null hypothesis is true, all populations are identical and have the same mean variance and shape