PSYCH: 42 (Chapter 12)
Independent Samples t Test
Used when two SEPARATE (or independent) samples are compared on some variable of interest
One-way b/w subjects ANOVA
Used when two or more samples are compared Two/Multiple independent samples Doesn't use a t distribution, but instead use an F distribution F distribution has properties that fairly diff from the t distribution
Interpretation of Post Hoc Test Results
With HSD = 1.94, this means that any difference between the groups must be at least 1.94 or larger to be statistically significant, This means for the three groups are as follows: M1 = 1 M2 = 4 M3 = 1 Differences between means: 1 and 2 [1-4] = 3 points 1 and 3 [1-1]= 0 points 2 and 3 [4-1] = 3 points (the absolute value are provided)
The numerator of the F-ratio measures the size of the sample mean differences. How is this number obtained, especially when there are more than two sample means?
by computing the variance for the sample means
An independent-measures t test produced a t statistic with df = 20. If the same data had been evaluated with an analysis of variance, what would be the df values for the F-ratio?
1, 20
When the null hypothesis is true for an ANOVA, what is the expected value for the F-ratio?
1.00
A research report concludes that there are significant differences among treatments, with "F(2,27) = 8.62, p < .01, η2 = 0.28." If the same number of participants was used in all of the treatment conditions, then how many individuals were in each treatment?
10
An analysis of variance is used to compare three treatment conditions with a group of 12 participants in each treatment. If a Scheffé test is used to evaluate the mean difference between the first two treatments, then what are the df values for the Scheffé F-ratio?
2, 33
An analysis of variance is used to evaluate the mean differences for a research study comparing four treatment conditions with a separate sample of n = 5 in each treatment. The analysis produces SS within treatments = 32, SS between treatments = 40, and SS total = 72. For this analysis, what is MS within treatments?
32/16
A researcher reports an F-ratio with df = 3, 36 for an independent-measures experiment. How many treatment conditions were compared in this experiment?
4
An analysis of variance produces SS between treatments = 40 and MS between treatments = 10. In this analysis, how many treatment conditions are being compared?
5
ANOVA Computational Approach
ANOVA takes the total variance and breaks it down into b/w groups variance and within groups variance Between groups variance measures the variability (spread) between the groups, and Within groups variance measures the variability (spread) within each group -how different within the group For ANOVA to be significant (for the groups to be declared as different) a high between groups variance and a low within group variance is desired
Step 3
Compute the Test Statistic F = MS between / MS within MS between --the variance (mean square) b/w the groups MS within -- the variance (mean square) within the groups
Step 4
Conclude and Write the Results in APA Format With the calculated F = 11.28 and a critical value = 3.38, the null hypothesis is rejected since 11.28 > 3.38 (it falls in the tail of the sampling distribution) Since the null hypothesis is rejected, we know that there is a difference somewhere between the three groups (we can't say at this point where the difference is) Written Results: The temperature of the room has a significant impact on learning, F (2,12) = 11.28, p < 0.05
How to use the F table
Degrees of freedom: Numerator -> this is the df between Degrees of freedom: Denominator -> this the df within F (2, 12) = 3.88 -with α = 0.05
To Obtain F
Following calculations need to be performed: SS Total = total variability as measured by SS SS Total = Σ x^2 - (G)^2 /N = 106 - (5+20+5)^2 / 15 = 46 SS Between = between variability as measured by SS SS Between = Σ (T^2/n - G^2/N) = (5^2/5 + 20^2/5 + 5^2/5 - 30^2/15) = 30 SS Within = within variability as measured by SS SS Within = SS1+SS2+SS2 = 6+6+4 = 16 As a check: SS Total = SS Between + SS Within 46 = 30 + 16 46 = 46 (confirms the calculations are correct) MS Between = between variance as measured by MS MS Between = SS Between / df Between = 30/2 = 15 MS Within = within variance as measured by MS MS Within = SS Within / df Within = 16/12 = 1.33 F = ration of between to within variance F = MS Between / MS Within = 15/1.33 F = 11.28
Calculating The Tukey's Post Hoc Test
In the Table, go to column 3 (k=3) and row 12 (df within=12), For alpha α = 0.05, q = 3.77 HSD = 3.77 (sqrt(1.33/5)) HSD = 1.94
Tukey's Post Hoc Test
Is listed as "HSD" in the text and the formula is: HSD = q (sqrt(MS within / n)) q --found in The Studentized Range Statistic (q) Table in Appendix textbook MS Within was calculated from the ANOVA and n= the sample size of a single group To find q, we need k ( the number of groups) and df for error term, which is equal to df within In our example, k=3 and df within = 12
F distribution
Is positively skewed and starts at zero One tailed
Chapter 12
One-way between subjects Analysis of Variance (ANOVA)
Post Hoc Tests
Post hoc testing --with post hoc tests, all pairs of groups are compared to see if there are any significant differences between the groups. For our example, the following post hoc tests will be conducted: --> only done when you are a significant ANOVA result (aka yes significance) 50° vs 70° 50° vs 90° 70° vs 90°
One-way b/w subjects ANOVA example
Psychologist examined learning performance under three temperature condition: 50 , 70, & 90 C Fifteen people were selected with five randomly assigned to each of the temperature condition (note that 5 people are in the 50 degree condition), 5 in the 70 degree condition, and 5 in the 90 degree condition) When different people are in different groups, this is known as "between-subjects"--hence the name One-way mean one independent variable, in this case the independent variable is temperature with three "levels or groups"--50, 70, & 90
ANOVA Source Table
Reference Table
Step 2
Set Alpha and Determine the Critical Value α = 0.05 (All ANOVA tests are one-tailed by design) There are two sets of degrees of freedom for the one-way b/w subjects ANOVA, df between and df within df between = k-1 df within = N-k df total = N-1 k= # of groups N= total # of people df between--with 3 groups, k -1 = 3-1 = 2 df = 2 df within--with 15 people total and 3 groups = N-k = 15-3 = 12 df = 12 Consult F table in Appendix of textbook df total = df between + df within = 2 + 12 = 14 df = 14
Step 1
State H0 and H1 H0: μ1 = μ2 = μ3 The population means are equal; there is no difference b/w the groups H1: At least one of the population means is different from the others -number of ways the null is false
Assumptions of one-way ANOVA
The assumptions for one-way ANOVA are identical to the assumptions for the intependent samples t test 1. The observations are independent 2. The scores are normally distributed in the population for each of the groups 3. The variances are equal in the population for each of the groups
Effect Sizes
The effect size for the one-way ANOVA is eta-square (n ^2) Formula on pic n^2 = 30/46 = 0.65 Effect Size Standards for eta-square are 0.01 , 0.06, 0.14 for small, medium, and large effect sizes Therefore, an eta-square of 0.65 corresponds to large (very large) effect size eta-square is also referred to as r^2 (not need to know r^2) only responsible to know n^2
Overall Results
The temperature of the room has a significant impact on learning, F(2, 12) = 11.28, p < 0.05. Tukey's post hoc test revealed that group 2 ( M = 4) was significantly higher than group 1 (M = 1) and group 3 (M = 1). There was not a significant difference between groups 1 & 3
Which of the following accurately describes the purpose of posttests?
They determine which treatments are different.
Post Hoc Test Written Results
With a minimum mean differences of 1.94 required for significance: groups 1 & 2 and groups 2 & 3 are significantly different while groups 1 & 3 are not significantly different. Group 2 (M = 4) is significantly higher than groups 1 and 3 (which each have M = 1). We'll add these findings to our final overall written results
Example Information
X1 (50 degrees) = 0, 1, 3, 1, 0 T1 = 5 SS1 = 6 n1 = 5 M = 1 X2 (70 degress) = 4, 3, 6, 3, 4 T2 = 20 SS2 = 6 n2 = 5 M2 = 4 X3 (90 degress) = 1, 2, 2, 0, 0 T3 = 5 SS3 = 4 n3 = 5 M3 = 1 T--sum of all x values Σ(X^2) = 106 G = 30 (sum of all Ts) N = 15 (total people in study) k = 3 (number of groups in study)
If an analysis of variance is used for the following data, what would be the effect of changing the value of SS2 to from 70 to 100?
increase SS within and decrease the size of the F-ratio
. ANOVA is necessary to evaluate mean differences among three or more treatments, in order to ____________________ .
minimize risk of Type I error
Under what circumstances are posttests necessary?
reject the null hypothesis with k > 2 treatments.
Which of the following is the correct description for a research study comparing problem-solving scores obtained for 3 different age groups of children?
single-factor design
. For an ANOVA, how does an increase in the sample sizes influence the likelihood of rejecting the null hypothesis and measures of effect size?
the likelihood of rejecting H0 will increase but there will be little or no effect on measures of effect size.
In an analysis of variance, the primary effect of large mean differences from one sample to another is to increase the value for ______.
the variance between treatments
In analysis of variance, an MS value is a measure of ______.
variance