ANOVAChapter 12

• Independent Variable: the variable manipulated by the experimenter in an effort to determine its effect

*ie. The “cause”

*note: there may be 2 or more levels of this variable

-an IV can be a condition of the subject’s external environment usually created by the experimenter

EX: exposure to violent tv, time of day, a drug, etc.

-an IV can be a characteristic of the subject

EX: gender, IQ, degree of attractiveness, etc.

**in this case the variable is not literally manipulated

**Instead, the experimenter assigns subjects to conditions based on the subject’s characteristics

• Dependent Variable: the behavior that may be affected by the independent variable.

*ie. The “effect”

*note: this is the variable that is measured

• Extraneous & Confounding Variables: influence the DV but not what the experimenter is studying.

-these must be controlled

-they cause error when they still influence the DV

**main source of this error is individual differences

**we would like all the subjects in our experiment to be exactly the same but that’s impossible

-therefore, we must have a measure of this error

EX: Standard Deviation measures error because it measures the differences between subjects

• t-test: used to test the effects of one IV with two levels to determine if the two groups (levels) are significantly different.

EXAMPLE: Are people affected in their judgments of guilt & innocence by how physically attractive the accused is?

M1= Attractive Defendant

M2=Unattractive Defendant

-the measure of error is the SED in the t-test

What if we wanted to include a third level of this one IV?

M1= Attractive

M2= Unattractive

M3= No information/Control Group

What if we wanted to test two IV’s?

Attractiveness of Defendant

Attractive Unattractive Control

Male

Gender of Defendant

Female

• The Type I error rate increases drastically with multiple comparisons

• The more statistical tests we perform, the more likely it is that some will be statistically significant purely by chance

• The following table illustrates how the alpha error originally set at .05 inflates as the number of decisions to reject increases:

• In doing research, we can not afford to take this kind of chance of committing alpha error.

-To correct for this, there is the Bonferroni Test and Analysis of Variance (ANOVA)

• Corrects for multiple t-tests (ie. multiple comparisons)

• Used to even out the level of significance among those t comparisons

-only used when comparisons are planned from the outset of the study

• Sets a lower alpha level for all the t tests combined

• Take the original alpha and divide it by the number of t tests performed

EX: If the original alpha is set at .05 and there are 3 t tests planned then the alpha is calculated:

α=.05/3=.016 or .02

• In this example, you would make three decisions about whether or not to reject Ho, each at the .02 level

• The theory behind the Bonferroni is that the probability of all the t-tests combined can not be greater than the sum of each individual t-test

EX: α=.016 + .016 + .016=.048 or .05

Ronald Fisher

• Compares more than two groups using only one statistical decision• Uses the F distribution so it’s also called “The F test”• “One-Way” when there is only one IV with various levels

Ho: μ1 = μ2 = μ3 =……μk k=the number of levels of the independent variableHa: μ1 ≠ μ2 ≠ μ3 ≠……μk Rejection of Ho tells us there is at least one significant difference but it doesn’t

tell us where it is

Logic of ANOVA Attractive Unattractive No Information x x x x x x

x x xx x xx x xm1 m2 m3

Investigates Three Sources of Deviation 1) Deviation of each score from the overall mean 2) Deviation of each group mean from the overall mean 3) Deviation of each score form its own group mean (measure of error)

Sum of Squares: sums of squared deviations from the mean-we square & sum the scores from each of the 3 sources of deviation to end up with 3 types of sums of squares

Developer of ANOVA

SStSSb

SSw

**NOTE:SSb + SSw = SSt

Overall Mean: M

SSt

SSb

SSw

Calculation of the Sums of Squares (SS):

• Variance Estimates: estimates of the population variance

-the sums of squares are just descriptive statistics

-In order to evaluate our data we must attempt to make inferences from the sample to population

-in order to make inferences, we calculate mean squares

Mean Squares: mean of the squared deviations

-ie. An independent estimate of the population variance

-SS is affected by sample size so we need to determine the mean of each source of deviation

-we use degrees of freedom to arrive at mean squares Total dft = N-1 MSb = SSb/dfb

Between dfb = k-1 Within dfw = N-k MSw = SSw/dfw

• F-ratio: the ratio of the difference between groups to the difference within groups

- Calculate:

• Use Table G: Critical values of F

-Use the df for the numerator and the df for the denominator to arrive at the critical values at the .05 or .01 levels

*df for the numerator = dfb

*df for the denominator = dfw

-On Table G, the lightface type is α=.05 and the boldface type is α=.01

-The obtained F is significant at a given level if it is equal to or greater than the critical value (table value)

**Note: If you have one IV with two levels t = √F and t2 = F

• Tukey’s HSD (Honestly Significant Difference) is a post-hoc multiple comparison test that can be applied when the null hypothesis is rejected.

-determines exactly where the significant differences are between groups

-we need critical values to evaluate the differences in the comparisons of an ANOVA

-Tukey’s HSD calculates the critical values

• Calculation:

Step 1: Determine the Tukey table value (Table H) at either α.05 or α.01

-use k and dfw (for the mean square within groups)

Step 2: plug the table value into the formula (in the place of α.05)

-plug all other values into the formula

Step 3: Find the differences between each pair of means (subtract them) A-B or M1 – M2

A-C or M1 – M3

B-C or M2 – M3

• Interpreting HSD

-the difference between any pair of means must be equal to or greater than Tukey’s HSD to reject Ho

• When there is only one IV with 3 or more levels, the ANOVA is a One-Way ANOVA

• When there is more than one IV, the ANOVA is a Factorial ANOVA

There are still 3 sources of deviation (with one change): 1) TOTAL: Deviation of each score from the overall mean 2) BETWEEN: -Difference between Male/Female: MAIN EFFECT A -Difference between Attractive/Unattractive/No Info: MAIN EFFECT B -Interaction of the 2 variables: A x B Interaction 3) WITHIN: Deviation of each score form its own group mean (measure of error)

• Example of a Three-Way ANOVA (2 x 2 x 3)

-3 Independent Variables

-7 F-Ratios will be evaluated

MAIN EFFECT A: Male vs. Female

MAIN EFFECT B: Property vs. Person Crime

MAIN EFFECT C: Attractive vs. Unattractive vs. No Info

A x B Interaction

A x C Interaction

B x C Interaction

A x B x C Interaction

• Before Calculation: Draw the block of cells

-One IV is set up in columns and the other IV is set up in rows

-a cell is where the row intersects the column

• Step 1: Calculate SSt and Identify C

• Step 2: Calculate SSb

• Step 3: Calculate SSw

• Step 4: Calculate SSA

• Step 5: Calculate SSB

• Step 6: Calculate SSAxB

C

• Step 7: Set up Summary Table, fill in SS & calculate df, MS, & F

• Step 8: Graph the Interaction

-start by calculating the means for each cell and the interactions

- IV #1 (or A or columns) is plotted on the X-axis in words

- DV is plotted on the Y-axis in numbers

- IV #2 (or B or rows) is represented by the lines that connected the plotted points

**use solid lines for one condition and a dashed line for the other

NOTE: If lines cross or they differ significantly from being parallel there is a significant interaction

• Step 9: Evaluate the 3 F-ratios for significance (Table G)

-same as with a one-way ANOVA but now you have three F-ratios instead of one

Degrees of Freedom (df) dfA= # of columns – 1 dfB = # of rows – 1 dfAxB = (dfA) (dfB) dfw = N-k

Mean Squares (MS) MSA= SSA/dfA MSB = SSB/dfB MSAxB = SSAxB/dfAxB

MSw = SSw/dfw

F MSA/MSw

MSB/MSw

MSAxB = MSw

SSSSA

SSB

SSAxB

SSw

• If in a two-way ANOVA an IV has more than 2 levels then calculation remains the same but evaluation changes slightly.

• The two main effects will have different df’s if the IV’s have a different number of levels

-in this case, you need to look up two sets of critical values in Table G

**Main effect A at the .01 and .05 levels

**Main effect B at the .01 and .05 levels

-You do not need to look anything up for the interaction

**if the F for the interaction is equal to or greater than ANY of the critical values you looked up for the main effects then you can reject the null