Learn to Use Factorial Analysis of Variance (ANOVA) in

Learn to Use Factorial Analysis of

Variance (ANOVA) in SPSS With

Data From the English Health

Survey (Teaching Dataset) (2002)

© 2019 SAGE Publications Ltd. All Rights Reserved.

This PDF has been generated from SAGE Research Methods Datasets.

Learn to Use Factorial Analysis of

Variance (ANOVA) in SPSS With

Data From the English Health

Survey (Teaching Dataset) (2002)

Student Guide

Introduction

This example dataset introduces Factorial Analysis of Variance (ANOVA). There

are a range of different types of ANOVA tests, including one-way and multiple

ANOVAs and MANOVAs. At their heart, all ANOVA tests examine the variance

of a continuous dependent variable(s) (for example, age, weight, and income)

across two (or more) levels of one or more independent categorical variable(s) (for

example, gender, class, and ethnicity). This example dataset introduces Factorial

ANOVA, which involves the ANOVA of a continuous dependent variable across

more than two levels of two or more categorical independent variables. This

example describes Factorial ANOVA, discusses the assumptions underlying it,

and shows how to compute and interpret it. We illustrate Factorial ANOVA using a

subset of data derived from the 2002 English Health Survey (Teaching Dataset).

Specifically, we test the extent to which the variance of an individual’s waist/

hip ratio differs by sex and marital status; does sex have more of an effect on

respondent’s waist/hip ratio (or vice versa) and is there a significant interaction

between sex and marital status in relation to respondent’s waist/hip ratio? This

page provides links to this sample dataset and a guide to producing a Factorial

ANOVA using statistical software.

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2019 SAGE Publications, Ltd. All Rights Reserved.

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Page 2 of 20 Learn to Use Factorial Analysis of Variance (ANOVA) in SPSS With Data

From the English Health Survey (Teaching Dataset) (2002)

What Is Factorial ANOVA?

There are a range of ANOVA tests; the simplest is a One-way ANOVA that

examines the variance of a continuous dependent variable across the

subgroupings of a categorical variable. Typically, Factorial ANOVA refers to an

ANOVA test that has more than two independent categorical variables (or

“factors”); sometimes it may also be referred to as a Two-way (two independent

variables) or Three-way (three independent variables) ANOVA. The inclusion of

more than three independent variables in an ANOVA is rarely done because

they become evermore difficult to interpret, and a multiple linear regression or

similar test would be easier to use. In Factorial ANOVA, variability is attributed

to differences both between groups and within groups; each level or subgrouping

and factor are paired up with each other. This allows us to identify interactions

between levels and factors; if an interaction is present, it means that the

differences in one factor depend on the differences in another factor. A Factorial

ANOVA seeks to compare the influence of at least two independent variables,

each with at least two levels, on a dependent variable. Our example examines

differences in variances in respondent’s waist/hip ratio (our dependent variable)

both within and between sex (male/female) and marital status (single/married-

cohabiting/previously married now single), our two independent variables. In a

One-way ANOVA, we would simply look at the difference in variances of

respondent’s waist/hip ratio between men and women. Our Factorial ANOVA

allows us to examine several issues: First, is sex the main effect on waist/hip ratio

or is marital status? Is there a significant interaction between the factors – how

do marital status and sex interact in relation to respondent’s waist/hip ratio and

can differences in sex and respondent’s waist/hip ratio be found in the different

levels of marital status. In our Factorial ANOVA, we can also look at the potential

interaction of marital status on sex differences, for example, might married men

have a greater mean waist/hip ratio than both married women and single men.

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When computing formal statistical tests, it is customary to define the null

hypothesis (H0) to be tested. In Factorial ANOVA, as with many more complex

statistical tests, there are three null hypotheses: H0a, which tests that there

will be no significant difference on respondent’s waist/hip ratio based on sex

(male/female); H0b, which tests that there will be no significant difference on

respondent’s waist/hip ratio based on marital status (single/married-cohabiting/

previously married now single); and H0c, which tests that there will be no

significant interaction between sex and marital status in terms of respondent’s

waist/hip ratio. Some difference is expected simply due to sampling error, i.e.,

random chance in sampling. The Factorial ANOVA conducted here will help us

determine whether the difference in the variances is large enough to have been

sufficiently unlikely to have arisen by chance alone, so that we may declare the

test statistically significant. “Large enough,” as usually defined, is a test statistic

with a level of statistical significance, or p-value, of less than .05. (That is, that a

difference in variances so large or larger would emerge only 5% of the time by

chance alone, under the maintained assumptions of the test.) This would lead us

to reject all or some of our null hypotheses and conclude that there likely is a

difference between the categories of the (independent) categorical variable and

the variances of the (dependent) variable being tested.

Calculating a Factorial ANOVA

Factorial ANOVA works by using the F-distribution, which is an asymmetric

distribution. The F-distribution is related to Chi-Square. In Factorial ANOVA, an

F-statistic is calculated by dividing the variance of the group means by the mean

of the within group variances. This F-statistic can then be compared to an F critical

value to determine whether to reject the null hypotheses (or not). If the F critical

value is larger than the F-statistic that your test has generated, then you can reject

the null hypothesis.

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To illustrate how to calculate, by hand, a Factorial ANOVA, let’s imagine that we

have collected some (hypothetical) data. We have collected data on the “level

of domestic contentment” (scored on a scale of 0–10, with 10 being extremely

contented and 0 being extremely discontented) expressed by heterosexual men

and women who are married/cohabit. We also collected data on their domestic

division of labour. Our dependent variable is “level of domestic contentment,”

which is continuous; our independent variables are “sex” (two levels: male/female)

and “domestic division of labour” (three levels: “I do the majority of the

housework,” “Partner does majority of the housework,” and “50/50 housework split

between partner and I”). Table 1 shows the data.

Table 1: “Level of Domestic Contentment”: “Sex” and “Domestic Division

of Labour.”

“I do the majority of the housework” “Partner the does majority of the housework” “50/50 split”

Men

3 5 7

4 3 8

5 3 7

3 5 8

4 5 7

1 5 7

Women

1 5 9

2 4 8

1 3 9

1 5 8

1 5 7

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2 4 9

We will test three null hypotheses:

H0a = There will be no difference on level of domestic contentment based

on sex

H0b = There will be no difference on level of domestic contentment based

on domestic division of labour

H0c = There will be no significant interaction between sex and domestic

division of labour in terms of level of domestic contentment

Social scientists generally choose a critical value for a Factorial ANOVA, such that

there is a less than a 0.05 probability that the result occurred strictly due to random

chance. Thus, researchers tend to reject the null hypothesis only when the F-test

statistic has a corresponding significance level (p-value) equal to or less than .05.

First, we need to calculate the degrees of freedom for our data. Table 2 shows this

data.

Table 2: Degrees of Freedom.

dfsex(A) a − 1 = 2 −1 = 1

dfdomesticdivision (B) b − 1 = 3 − 1 = 2

dfsex × dfdomesticdivision(A × B) (2−1)(3−1) = 2

dferror N − ab = 36 − (2 × 3) = 30

dfsex(A) N − 1 = 36 − 1 = 35

Now that we have our hypotheses and df, we should now locate the F critical

value for each of our hypotheses by consulting a table of the critical values

of the F-distribution. These are available in the appendices of most statistics

textbooks and can be found online easily. Table 3 shows the critical values for

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each hypothesis.

Table 3: Critical Values of F.

Hypothesis df Critical value

of F

H0a: There will be no difference on level of domestic contentment based on sex. 1,

30 4.17

H0b: There will be no difference on level of domestic contentment based on domestic division of

labour.

2,

30 3.32

H0c: There will be no significant interaction between sex and domestic division of labour in terms of

level of domestic contentment.

2,

30 3.32

We can now calculate our test statistic F. To do this, we need to do a number

of calculations; firstly, we need to calculate the sum of squares (SS) for each

hypothesis, the error, and the total. We can do this using Equation 1.

(1)

SSsex =∑ (∑ ai )

2

(b)(n)−

T2

N

Let’s calculate SSsex to illustrate.

SSsex =932 + 842

(3)(6)−

1772

36

= 2.25

We can now calculate the other SS using versions of the same equation.

SSdomesticdivision =302 + 532 + 942

(2)(6)−

1772

36

= 175.17

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SSsex x domestic division =∑ (∑ ai bi)

2

(n)−

∑ (∑ ai )2

(b)(n)−

∑ (∑ bi )2

(a)(n)+

T2

N

SSsex x domestic division =222 + 82 + 272 + 262 + 442 + 502

(6)−

932 + 842

(3)(6)−

302 + 532

(2)(6

= 17.16

SStotal = ∑ Y2 −1772

36

SStotal = 1081 −1772

36= 210.75

SSerror = SStotal − ( SSsex + SSdomesticdivision + SSsex x domestic division)

= 16.17

Next, we need to calculate our mean squares (MS) which we can do using

Equation 2.

(2)

MS =SSdf

Table 4 shows all our calculations (inlcuding MS).

Table 4: Calculations for SS, df, and MS.

SS df MS

Sex 2.25 1 2.25

Domestic division of labour 175.17 2 87.59

Interaction 17.16 2 8.58

Error 16.17 30 0.54

Total 210.75 35

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Now we can calculate our F test statistic, using Equation 3

(3)

F =MSeffect

MSerror

Fsex =2.250.54

= 4.166

Fdomesticdivision =87.590.54

= 162.20

Fdomesticdivision =87.590.54

= 162.20

Now that we have calculated our F test statistic and we already know our F critical

values, we can return to our null hypotheses. For H0a there will be no difference

on level of domestic contentment based on sex, our test statistic is F = 4.166,

which is less than the critical value F (4.17), and so we fail to reject the null

hypothesis. For H0b there will be no difference on level of domestic contentment

based on domestic division of labour, our test statistic is F = 162.20, which is

greater than the critical value F (3.32), and so we reject the null hypothesis. For

H0c there will be no significant interaction between sex and domestic division

of labour in terms of level of domestic contentment, our test statistic is F =

15.89, which is greater than the critical value F (3.32), and so we reject the null

hypothesis.

We can summarise our findings by stating that there is no sex difference in

terms of level of domestic contentment but that there were significant differences

between the three different levels of domestic division of labour. In addition, an

interaction effect was present.

This example has shown you how to calculate an F-statistic manually, which is

quite cumbrous, and only really feasible to do with small samples. Especially with

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http://methods.sagepub.com/dataset/factorial-analysis-variance-hse-2002#i85

larger samples, it is much easier to use statistical software.

Assumptions Behind the Method

All statistical tests rely on some underlying assumptions, and they all are affected

by the type of data that you have.

Assumptions of Factorial ANOVA:

• The dependent variable must be continuous.

• There must be two or more independent variables indicating two or more

categorical groupings that partition the sample.

• There must be independence of observations, so there is no relationship

between the groups or between the observations in each group.

• The data must be normally distributed or large enough for satisfactory

asymptotic approximation.

• The data must have equality of variance across factors and levels.

Assumptions 1, 2, and 3 are not typically testable from the sample data and are

related to the research design. The third assumption is only likely to be violated if

the data were sampled by pairs rather than individuals (e.g., couples rather than

individual persons). It is important to understand how your data were collected

and categorised, this will help you avoid violating the first three assumptions. The

fourth and fifth assumptions can be easily tested using statistical software.

Illustrative Example: Sex, Marital Status, and Respondent’s Waist/Hip

Ratio

This example presents a Factorial ANOVA using three variables from the 2002

English Health Survey (Teaching Dataset). Specifically, we test the extent to which

the variance of an individual’s waist/hip ratio differs by sex and/or marital status;

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and is there a significant interaction between sex and marital status in relation to

waist/hip ratio

Thus, this example addresses the following research question:

Does an individual’s waist/hip ratio vary by sex and marital status?

Our model has three null hypotheses:

H0a: There will be no difference in variance of waist/hip ratio based on

sex.

H0b: There will be no difference in variance of waist/hip ratio based on

marital status.

H0c: There will be no significant interaction between sex and marital

status in terms of waist/hip ratio.

The Data

This example uses a subset of data from the 2002 English Health Survey

(Teaching Dataset). This extract includes 5,123 adult respondents. Please note

that the original dataset is larger than this, but it has been “cleaned” to reduce its

size for ease of analysis and also to include only those who have responded to

our dependent variable. The three variables we examine are:

• Respondent’s sex (sex)

• Marital status including cohabitees (RECODE) (RECODEmarstab)

• Valid Mean Waist/Hip ratio (whval)

The first variable, sex, is coded 1 if a respondent is “male” and 2 if “female.”

The second variable, RECODEmarstab, is coded 1 if respondent is “single,”

2 if “married/cohabiting,” and 3 if “previously married now single.” Given that

our dependent variable, whval, is continuous and our independent variables are

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categorical with two or more groups each, then Factorial ANOVA is appropriate for

these data.

Analysing the Data

Before conducting Factorial ANOVA, we should first examine each variable in

isolation; this is called univariate analysis. We start by presenting a frequency

table of sex (Table 5). We can see that there are slightly more females (54.1) than

makes (45.9%).

Table 5: Frequency Distribution of sex.

Frequency Percent Valid percent Cumulative percent

Valid

Male 2,354 45.9 45.9 45.9

Female 2,769 54.1 54.1 100.0

Total 5,123 100.0

Table 6 below shows the frequency distribution of RECODEmarstab. We can see

that the majority (66.4%) of respondents are married/cohabiting, while only 14.9%

are “previously married now single.”

Table 6: Frequency Distribution of RECODEmarstab.

Frequency Percent Valid percent Cumulative percent

Valid

Single 958 18.7 18.7 18.7

Married/cohabiting 3,402 66.4 66.4 85.1

Previously married now single 763 14.9 14.9 100.0

Total 5,123 100.0 100.0

Table 7 shows the frequency distribution of whval. The mean waist/hip ratio is

0.86, which is deemed a moderate health risk for women but low health risk for

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men.

Table 7: Frequency Distribution of whval.

whval

N

Valid 5,123

Missing 0

Mean 0.8622

Median 0.8581

Standard deviation 0.08718

Range 0.66

Minimum 0.57

Maximum 1.24

Tables 5, 6, and 7 show the distribution of the variables individually, but our

interest lies with their relationship with each other.

Prior to running our Factorial ANOVA, we need to test our data for normality and

homogeneity. We can do this in a variety of ways, but a good way to test for

normality is to run Q:Q Plots on statistical software, such as IBM® SPSS®. Figure

1 shows the Q:Q Plots for our data.

Figure 1: Q:Q Plots of sex: whval.

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Figure 2: Q:Q Plots of RECODEmarstab: whval.

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Figures 1 and 2 show that our data are approximately normal. We can test for

homogeneity using Levene’s test of Homogeneity of Variance (Table 8).

Table 8: Levene’s Test of Homogeneity of Variance.

whval sex RECODEmarstab

Levene statistic (based on mean) 2.981 8.245

df 1 2

df2 5,121 5,120

Sig. 0.84 0.000

The results of the Levene’s test suggest that our data’s homogeneity is affected

by the large subgrouping of “married/cohabiting” within the RECODEmarstab

variable, as we have homogeneity between sex:wvhal. However, if we review the

variance of wvhal within RECODEmarstab, we can see that the variances are very

similar (Table 9).

Table 9: Variance of wvhal: RECODEmarstab.

Variance of wvhal

Single 0.007

Married/Cohabiting 0.008

Previously married, now single 0.006

Given that the Levene’s test can be easily skewed and our variances look similar,

we will proceed with the Factorial ANOVA.

Conducting Factorial ANOVA

Table 10 presents the findings of our Factorial ANOVA. We can see that our model

is statistically significant (F = 776.436, p = .00), indeed the Adjusted R Squared

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(0.431) suggests that sex and marital status account for 43% of the variance in

waist/hip ratio in this model.

Table 10: Results Factorial Analysis of Variance.

Tests of between-subjects effects

Type III sum of squares df Mean square F Sig.

Corrected model 16.792* 5 3.358 776.436 .000

Intercept 2,297.615 1 2,297.615 531,189.947 .000

RECODEmarstab 1.670 2 .835 193.009 .000

Sex 8.149 1 8.149 1,884.023 .000

RECODEmarstab*sex .213 2 .106 24.585 .000

Error 22.133 5,117 .004

Total 3,847.389 5,123

Corrected Total 38.925 5,122

* R Squared = 0.431 (adjusted R Squared = 0.431).

From Table 10 we can see that when we control for marital status, there is a

difference in respondent’s waist/hip ratio by sex (F = 1,884.023, p = .000), causing

us to reject H0a that there will be no difference in variance of waist/hip ratio based

on sex. Likewise when we control for sex, there is a difference in respondent’s

waist/hip ratio by marital status (F = 193.009, p = .000), causing us to reject H0b

there will be no difference in variance of waist/hip ratio based on marital status.

Finally, if we can reject (F = 24.585, p = .000) H0c there will be no significant

interaction between sex and marital status in terms of waist/hip ratio. Figure 3

confirms this rejection of H0c.

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http://methods.sagepub.com/dataset/factorial-analysis-variance-hse-2002#i122

Figure 3: Graph of the Interaction Effect of Sex-Marital Status on

Waist/Hip Ratio.

Overall, we can reject all three null hypotheses. Our findings suggest that

respondent’s waist/hip ratio varies by sex and marital status, even when we

control for each; relatedly, there is a significant interaction between sex and

marital status on respondent’s waist/hip ratio.

Further testing of these data, such as analysis of post hoc tests or calculation of

effect sizes would generate further detail as to the impact of specific subgroupings

within variables on waist/hip ratios.

Presenting Results

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A Factorial ANOVA can be reported as follows:

“A Factorial ANOVA was conducted to compare the main effects of sex and marital

status and interaction between sex-marital status on respondent’s waist/hip ratio.

We used a subset of the 2002 English Health Survey (Teaching Dataset).

H0a = There will be no difference in variance of waist/hip ratio based on

sex.

H0b = There will be no difference in variance of waist/hip ratio based on

marital status.

H0c = There will be no significant interaction between sex and marital

status in terms of waist/hip ratio.

This extract includes 5,123 adult respondents. A Two-way ANOVA was conducted

on the influence of two independent variables (sex, marital status) on respondent’s

waist/hip ratio. Sex included two levels (male, female) and marital status consisted

of three levels (single, married/cohabiting, previously married now single). All

effects were statistically significant at the 0.05 significance level. The main effect

for sex yielded F (1, 5117) = 1,884.023, p = .000, indicating a significant difference

between males (M = 0.9205, SD = 0.07200) and females (M = 0.8126, SD =

0.0655). The main effect for marital status yielded F (2, 5117) = 24.585, p = .000,

indicating a significant difference between single (M = 0.8298, SD = 0.08330),

married/cohabiting (M = 0.8709, SD = 0.08776) and previously married now single

(M = 0.8642, SD = 0.06699). The interaction effect was significant, F (2, 5117) =

24.585, p = .000.”

Review

Factorial ANOVA is a test that seeks to compare the influence of at least two

independent variables, each with at least two levels, on a dependent variable. It

tests the null hypotheses of no difference in variances between and within the

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groups.

You should know:

• What types of variables are suited for Factorial ANOVA.

• The basic assumptions underlying this statistical test.

• How to compute and interpret Factorial ANOVA.

• How to report the results of a Factorial ANOVA.

Your Turn

You can download this sample dataset along with a guide showing how to produce

a Factorial ANOVA using statistical software. The sample dataset also includes

another variable called saywgt, which is respondent’s view of own weight. See

whether you can reproduce the results presented here for the sex and

RECODEmarstab variables and then try producing your own Factorial analysis

substituting saywgt for sex in the analysis.

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Documents

Learn to Use Factorial Analysis of Variance (ANOVA) in