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Purpose IV DV Inferential Test

Test if sample representative of the population

X Continuous One Sample Case Z-test (μ and σ known)

Test if sample representative of the population

X Continuous One Sample Case t-test (σ unknown)

Test differences Dichotomous Continuous Independent (or dependent) t-test

Test differences Categorical (2+ groups)

Continuous One-Way ANOVA

Test relationship Continuous Continuous Pearson Correlation

Test relationship Dichotomous Continuous Point Biserial

Test if observed frequencies fit expected frequencies

Categorical Categorical Chi-Square Test of Goodness-of-Fit

Test relationship Categorical Categorical Chi-Square Test of Independence/Association

Where are we and where are we going?

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Week 12: One-way Analysis of Variance (ANOVA)

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One-Way ANOVA Definition

A One-Way ANOVA is used when comparing two or more group means on a continuous dependent variable.

The independent t-test is a special case of the One-Way ANOVA for situations where there are only two group means.

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One-Way ANOVA Definition Con’t.

Therefore, if there are only two groups, an Independent t-test is a shortcut to using a One-Way ANOVA, but provides the same results. It is an extension of Independent t-test.

For situations involving more than two groups, a One-Way ANOVA must be used.

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Types of Variables for One-Way ANOVA

The IV is categorical. This categorical IV can be dichotomous (i.e., two groups) or it can have more than two groups (i.e., three or more groups).

The DV is continuous

Data are collected on both variables for each person in the study.

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Examples of Research Questions for One-Way ANOVA

Is there a significant difference among freshmen, sophomores, and juniors on college GPA? OR Is there a significant difference in GPA by class level?Note that GPA is continuous and class standing is categorical (freshmen, sophomores, or juniors)

Is there a significant difference in student attitudes toward the course between students who pass or fail a course?Note that student attitude is continuous and passing a course is dichotomous (pass/fail). Because the IV has only two groups, an independent t-test or rpb could also be used here.

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Examples of Research Questions for One-Way ANOVA Does student satisfaction significantly differ by location

of institution (rural, urban, suburban)?Note that student satisfaction is continuous and institution location is categorical.

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More Research Questions

Is there a significant difference in weight loss when dieting, exercising, and dieting and exercising?

Is there a significant difference in child’s self-esteem by parenting style (authoritative, permissive, and authoritarian)?

Is there a significant difference among those who condition, blow-dry, or only shampoo in hair frizziness?

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Why not do multiple t-tests?

Doing separate t-tests inflates Type I error rate.

An ANOVA controls the overall error by testing all group means against each other at once, so your alpha remains at, for instance .05.

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Why not do multiple t-tests?

Any experiment with k groups has k X (k-1)/2 different pairs available for testing.

k = 3 k X (3-1) / 2 = k X 3 (2/2) = 3(1) = 3 different pairs available for testing

We need to compare:Sample 1 vs. Sample 2Sample 1 vs. Sample 3Sample 2 vs. Sample 3

So we need to perform 3 t-tests!

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Type I error rate inflate:

Sample 1 vs. Sample 2 α = .05 Sample 2 vs. Sample 3 α = .05 Sample 2 vs. Sample 3 α = .05

If alpha = .05 for each test, times 3 tests, the new probability of a Type I error is about 0.15 now!

Type I error rate: 1-(1-α)c

Whereby: α = alpha for each separate t-test

c = number of t-tests

Type I Error Rate Inflate

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Type I error rate inflate:Type I Error Rate Table

Retrieved from espse.ed.psu.edu/edpsych

Error rates with repeated t-tests

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Type I error rate inflate:Assumptions

The populations from which the samples were obtained must be normally or approximately normally distributed.

The samples must be independent.

The variances of the populations must be equal.

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Major Concepts: Calculating Sums of Squares

The One-Way ANOVA separates the total variance (hence the term – analysis of variance) in the continuous dependent variable into two components: variability between the groups and variability within the groups.

Variability between the groups is calculated by first obtaining the sums of squares between groups (SSb), or the sum of the squared differences between each individual group mean from the grand mean.

Variability within the groups is calculated by first obtaining the sums of squares within groups (SSw) or the sum of the squared differences between each individual score and that individual’s group mean.

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Major Concepts: Calculating Sums of Squares

The linear model, conceptually, is: SSt = SSb + SSw, where SSt is the total sums of squares.Note the simplicity of statistical models: They are linear depictions of phenomena, splitting variability into (1) what we can measure systematically (differences between groups) and (2) what we can’t explain or account for systematically (differences within groups).

Variability within the groups is random variance or noise; it reflects individual differences from the group mean and sometimes prevents us from seeing difference between groups.

Variability between the groups is systematic variance, it reflects differences among the groups due to the experimental treatment or characteristics of group membership.

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Major Concepts: One-Way ANOVA as a ratio of variances Formula for variance:

We can see that the numerator is a sum of squared values (or a sums of squares), and the denominator is the degrees of freedom. Thus, the formula can be re-written as:

1 )x -x ( 2

2

N

dfSS2

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Major Concepts: Variance Components in ANOVA

The Analysis of Variance (ANOVA) analyzes the ratio of the variance between groups (i.e., how far apart the group means are from one another) to the variance within the groups (i.e., how much variability there is among the scores within a single group).

In ANOVA, these variances, formerly known to us as σ2, are referred as mean squares (MS) or the average of the sums of squares (SS/df).

Thus, a mean square between is simply the variance between groups obtained by a sums of squares divided by degrees of freedom (SSb/dfb).

Likewise, a mean square within is simply the variance within the groups also obtained by a sums of squares divided by degrees of freedom (SSw/dfw). Both are shown below:

b

bb dfSSMS

w

wdf

SSMSw

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Table 1. Scores and means of the comparison groups and the total group

1 2 3

1 8 7

4 6 6

3 7 4

2 4 9

5 3 8

1 5 5

6 7

5

7 6 8

3.14 5.50 6.38

n

mean

Method

Within

Between

Within

Within

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Partitioning the Variance

Between

Within

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Major Concepts: Factors that Affect Significance

The MSb and the MSw are then divided to obtain the F ratio for hypothesis testing, i.e., F is the ratio of MSb and the MSw

As in all statistical tests, the larger the numerator (for example, the larger the difference between groups), the larger the test statistic (whether it be r, t, or F), and the more likely we are to reject the null hypothesis.

Also, the smaller the denominator (the less variability among people in a group), the larger the test statistic, and the more likely we are to reject the null hypothesis.

w

bMS

MSF

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Major Concepts: Factors that Affect Significance

The diagrams below show the impact of increasing the numerator of the test statistic. Note that the within group variability (the denominator of the equation) is the same in situations A and B. However, the between group variability is greater in A than it is in B. This means that the F ratio for A will be larger than for B, and thus is more likely to be significant.

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Major Concepts: Factors that Affect Significance

The diagrams below show the impact of decreasing the denominator of the test statistic. Note that the between group variability (the difference between group means) is the same in situations C and D. However, the within group variability is greater in D than it is in C. This means that the F ratio for C will be larger than for D, and thus is more likely to be significant.

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Major Concepts: t2 = F

Diagrams A through D on the previous slides could also be used to illustrate the t statistic. As two groups differ more in their means (increase in numerator and effect size), the larger the value of t and the more likely we are to find significant results.

The lower the within-group variability (decrease in noise), the smaller the denominator and the more likely we are to find significance. This analogy can be made to ANOVA because t is just a special case of ANOVA when only two groups comprise the independent variable.

We are familiar with the t distribution as normally distributed (for large df), with positive and negative values. The F statistic, on the other hand, is positively skewed, and is comprised of squared values. Thus, for any two group situation, t2 = F.

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Distribution of F ratio

F distribution is positively skewed.

If F statistic falls near 1.0, then most likely the null is true.

If F statistic is large, expect null is false. Thus, significant F ratios will be in the tail of the F distribution.

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SPSS Example of One-Way ANOVA

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Summary of Calculations

dfb groups = k – 1 where k is # of groups dfw groups = N – k where N is total # of individuals in groups

MSb= SSb/ dfb

MSw = SSw/ dfw

F = MSb/ MSw

*SS is the sum of the squared deviations otherwise denoted S2

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Steps in One-Way ANOVA

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Step 1: State Hypotheses

There is no significant difference among the groups in variable x.

μ1 = μ2 = μ3 = μ4… There is a significant difference between

at least two of the groups in variable x. In other words, at least one mean will significantly differ.

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Step 2: Set the Criterion for Rejecting Ho

“Between” (k-1)

“Within”

(N – K)

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Practice with Table C.5, p. 640

3 groups 30 people df between = k – 1 = 3 – 1 = 2 df within = N – k = 30 – 3 = 27What is my critical value? Remember, if the

actual value isn’t in the table, use the next lowest value to be conservative.

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Step 3: Compute Test Statistic

1. Calculate Correction Factor:

(T)2 total scores2

N total number of scores

CF= (sum of each group)2  / total number of observations.

2. Calculate SSB: k = # of groups; n = number of scores

SSB = ∑ TK2 - Correction factor

nk

Sum the total for each group and square / n . Subtract correction factor.

3. Calculate SST: Square each individual score. Sum the squared scores across all groups and subtract correction factor.

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Step 3: Compute Test Statistic

4. Calculate SS within:

SSW = SST – SSB

5. Calculate MSB and MSW

6. Calculate the F ratio F =

-------

dfSSMS

B

BB

dfSSMS

W

WW

MS B

MSW

N - KK-1

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Step 4: Compare Test Statistic to Criterion

Like t, a large F value indicates the difference (or treatment effect) is unlikely due to chance.

When F-ratio is close to 1.0, it is likely that the Ho is true.

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Step 5: Make Decision

Fail to reject the null hypothesis and conclude that there is no significant difference among the groups F(dfb,dfw) = insert F statistic, p > insert α.

Reject the null hypothesis and conclude that there is a significant difference among the groups F(dfb,dfw) = insert F statistic, p < insert α.

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Interpreting Tables

Treatment A Treatment B Treatment C

M 2.5 6.0 0.5

SE .43 .58 0.22

Source SS df MS F

Between treatments 93 2 46.5 41.02

Within treatments 17 15 1.13

Total 110 17

ANOVA SummaryK - 1 N - K

N - 1

SSb/dfb

SSw/dfw

MSb/MSw

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Suggested Study Cards

Assumptions of ANOVA. EQUATION FOR ANOVA (slide 16 for

components). Equation for Type I error rate (alpha

inflation). Equation for different number of pairs

available for testing.