Exam #6 (chapter 13: ANOVA)
n is the number of participants in each group ΣX is the sum of scores in each group X is the mean of each group Σ(X2) is the sum of each score squared (ΣX)2 / n is the sum of scores squared divided by n (the sum of squares (SS) for that group
Computing the F Statistic: What do we need for each group?
ΣΣX = ΣX + ΣX + ΣX (ΣΣX)2 / N (remember N = n + n + n) ΣΣ(X2) = add all of them Σ(ΣX)2/n = add all of them
Computing the F Statistic: What do we need to do to calculate all three groups altogether?
N is the TOTAL number of participants ΣΣX is the sum of all scores across groups (ΣΣX)2/N is sum of all scores squared divided by N ΣΣ(X2) is the sum of all the sums squared Σ(ΣX)2 /n is the sum of the sum of each group's scores squared and divided by n (the sum of the ss for each group)
Computing the F Statistic: what do we need for ACROSS group calculations?
Between groups: k-1 k = the number of groups Within groups: N-k N = the total sample size, or 30 in our case k = the number of groups
Degrees of Freedom Formulas
"We ran a one way ANOVA with condition as our IV (high vs. low vs. control) and amount participants recalled spending on books as our DV. The test was significant, F(2, 27) = 7.15, p < .01. Tukey post hoc tests showed that participants recalled spending more money on books if they saw "fake" prior participants spending high amounts (M = $197, SD = ) than if they saw "fake" prior participants spending low amounts (M = $172, SD = ). However, neither the high nor low conditions differed from the control condition (M = $185, SD = ) in their recall."
Example of writing up the ONE Way ANOVA
Recall that we allow some error in our research design (we use p < .05 rather than saying we need to be 100% sure our IV was the only thing to influence the DV) Running multiple t-Tests makes it likely that at least one of our multiple t-Tests will be significant. This increases our Type I error (saying that something is significant when it really isn't)
How can running three t-Tests be cheating?
We want the F (our obtained value) to be high enough to overcome a critical value (from an F table). Logically, the bigger the MS between number and the smaller the MS within, the higher the F will be
How do we want our obtained and critical value to be in the F-test?
F Table 1. We have a df column for the denominator (N - k, or total participants minus # of IV levels) 2. We have a df for the numerator (k - 1) The numerator is the top part of the equation (MS between) and the denominator is the bottom of the equation (MS within) 3. In the table, we can also find our error rate (the level of risk we are willing to set) the values for p < .01, p < .05, and p < .10
How do you determine the value needed to reject the null?
ONE independent variable, however, it happens to have three or more levels We could have one experimental group and two control groups, two experimental groups and one control group, or three experimental groups.
How many Independent Variables will you see in an ANOVA?
Null hypothesis: Groups do not differ, Ho: µ1 = µ2 = µ2 Alternative hypothesis: Groups differ, H1: X1 ≠ X2 ≠ X3 No need for a one-tailed versus two-tailed ANOVA. An F test actually looks at all comparisons and all differences between all means—it's an "omnibus" test
How would you state the null and alternative hypothesis in a F Statistic Test?
1.49 ( 8.71 / 5.87 = 1.49 )
If I MS within is 5.87 and MS between is 8.76, what is your F value?
n = 30, N = 90
If I have 3 groups of 30 people each, what is my n and what is my N?
144 ( 3+ 4 + 5 = 12^2 = 144 )
If my values are 3, 4, and 5, what is the (ΣX)2?
50 ( 3^2 + 4^2 + 5^2 = 50 )
If my values are 3, 4, and 5, what is the Σ(X2)?
Two; One; Three or more
Independent t-Tests involve ___ groups; dependent t-Tests involve ___ groups; ANOVAs involve ___ groups.
MS between = SS between / df between MS within = SS within / df within
MS Formulas
Post hoc tests
Since ANOVAs have three or more levels to the IV, we need to run additional tests called ___ ___ ___ to fully figure out which of the three or more means differ
SS between = Σ (Σx)2/n -(ΣΣx)2/N SS within = ΣΣ(x2) - Σ (Σx)2/n
Sum of Squares Formulas
Higher; lower
The ___ the between group variability (or MS between) and the ___ the within variability or (MS within), the higher the F.
Between group; within group
To get a large F-value, ___ should be a high number and ___ should be a low number.
It is actually GOOD variability, as it shows that there is a difference between our groups Since we manipulated our independent variable, we really want there to be differences between IV levels
Variance that is due to differences between groups means what exactly?
This is BAD variability, as it relates to variability not due to the presence of the independent variable Sources of within group variability involve a wide variety of "error", like experimenter bias, sampling error, measurement error, participant differences. Thus it is ANYTHING other than the IV that affects the dependent variable
Variance that is due to differences within groups means what exactly?
We WANT to increase between group variability and decrease within group variability
We WANT the F to be high for what reasons?
If we run a single t-Test, our error is 5% (p < .05) If we run two t-Tests, our error is 10% Three t-Tests drives that error up to nearly 15%! It's a lot easier to find significance with 15% probability of occurring (15% error) than it is with 5% The ANOVA runs just ONE test
What are the statistical reasons behind not running multiple t-Tests?
What these tests do is compare all combinations of the three or more IV levels at the same time. It compares A and B, B and C, and A and C
What do Post hocs, such as LSD, Tukey, Duncan, S-N-K, etc. do?
Balance the between variability and within variability Formula: F = MS between / MS within
What do you try to do with the ANOVA?
The major difference here is that instead of looking at only two means, we compare and contrast three or more means/ We are comparing across all three groups, rather than between just two groups.
What is the major difference between the ANOVA and the t-Test?
Our job is to compare our obtained value (F = 7.15) to the critical tabled values given our numerator df = 2 and our denominator df = 27 Using the F table, find the df for the numerator in the top row (2) and the df for the denominator in the first column (27). That is, go across 2 columns and down 27 rows.
What should you do once you get your obtained and critical value?
1. We are testing for a difference between scores of different groups 2. The "real" participants are only tested once 3. There are at least three levels 4. The correct test is the simple Analysis Of Variance
When do we use the ANOVA?
F(4, 85)
You run a study with 90 participants who participate in one of the five levels of the independent variable. How would you write up your df in a results section?
4 ( k-1 = 5-1= 4 )
You run a study with 90 participants who participate in one of the five levels of the independent variable. What is your BETWEEN group degrees of freedom?
85 ( N-k = 90-5 = 85 )
You run a study with 90 participants who participate in one of the five levels of the independent variable. What is your WITHIN group degrees of freedom?
