Statistics for Marketing and Consumer Research

Analysis of variance

Chapter 7

Statistics for Marketing & Consumer Research

Copyright © 2008 - Mario Mazzocchi

1

Tests on multiple hypotheses

• Consider the situation where the means for more than two groups are compared, e.g. mean alcohol expenditure for:

(a) students; (b) unemployed; (c) employees

• One could run a set of two mean comparison tests (students vs. unemployed, students vs. employed, employed vs. unemployed)

• However, as seen in lecture 6, each of these tests is subject to Type one error (the level of significance a ), i.e. the probability of rejecting the null hypothesis when it is actually true

• Thus, the overall Type one error for the joint three tests is larger than a because the probability of Type one error increases with the number of tests

• This is the so-called problem of inflated family-wise (or experiment-wise) Type one error



2

Analysis of Variance

• It is an alternative approach to mean comparison for multiple groups

• It is a set of techniques for a variety of situations

• It is applicable to a sample of individuals that differ for one or more given factors

• It allows tests where variability in a variable is attributable to one (or more) factors



3

Example

EFS: Total

Alcoholic

Beverages,

Tobacco

Mean

St. Dev.

Self-





Economic position of Household Reference Person employed

Fulltime employee





Pt employee





Unempl.





Ret unoc over min ni age





Unoc - under min ni age TOTAL







18.56





14.64





12.39





19.48







7.34



11.99





12.67



19.0 18.5 15.0 19.7 14.6 19.1 17.8

Are there significant difference across the means of these groups?

Or do the differences depend on the different levels of variability across the groups?



4

Analysis of Variance

• Here the target variable is alcohol expenditure, the factor is the economic position of the HRP

• Basically we investigate the attribution of a variation in the metric target variable to variations in one on more categorical explanatory variables (the factors)

• A treatment is a combination of different factors in n-way analysis of variance



5

One-way ANOVA

• Only one categorical variable (a single factor)

• Several levels (categories) for that factor

• The typical hypothesis tested through ANOVA is that the factor is irrelevant to explain differences in the target variable (i.e. the means are equal, as in bivariate mean comparisons/t-tests)

• Apart from the tested factor(s), the groups should be safely considered homogeneous between each other



6

Null and alternative hypothesis for

ANOVA

• Null hypothesis (H

0

): all the means are equal

• Alternative hypothesis (H

1 means are different

): at least two



7

Arranging data for ANOVA

Overall mean

1

Self-

Economic position of Household Reference Person

Group (g)

2 3 4 5 employed

Fulltime employee

Pt t employee

Unempl.

Ret unoc over min ni age

6

Unoc - under min ni age x

11 x

21 x

31

… n

1 x

1 x

21 x

22 x

32

…

Observations x

13 x

23 x

33

… x

14 x

24 x

34

… x n

2

Number of observations ( n ) n

3

Means n

4 x

2 x

3 x

4 x

15 x

25 x

35

… n

5 x

5 x

16 x

26 x

36

… x n

6

6



8

The statistical distribution to carry out

ANOVA

1. Decompose the total variation (sum of squares corrected for the mean)

2. Compute the F-test statistic

3. Choose the critical value

4. Interpret the result



9

One-way ANOVA: data

• Suppose that we have n observation within each group and g group

Obs.

1

2

… i

… n

Group mean

TOTAL MEAN

Group (factor level)

1 2 … j … g x

11 x

12

… x

1j

… x

1g x

21 x

22

… x

2j

… x

2g

… … … … … … x i1 x i2

… x ij

… x ig

… … … … … … x n1 x n2

… x nj

… x nn x

1 x

2

… x j

… x g x



1 g  g j



1 x j

10



Measuring and decomposing the total variation

• SUM OF SQUARES (corrected for the mean)

VARIATION BETWEEN THE GROUPS +

VARIATION WITHIN EACH GROUP=

________________________________

TOTAL VARIATION



11

Variance decomposition

s

T

2  g n c  c



1 r



1

 x rc

 x



2 n



1

(TOTAL VARIANCE)

2 s

BW

 g  c



1

 x c

 x



2 n c g



1

2 s

W

 g n c  c



1 r



1

 x rc

 x c n c



1



2

(VARIANCE BETWEEN GROUPS)

(VARIANCE WITHIN GROUPS) g n c  c



1 r



1

 x rc

 x



2  g n r  c



1 r



1

 x rc

 x c



2  c g 



1

  

2 x x n c c

1





1

( n 1) n g



12

The basic principle of the ANOVA

• If the variation explained by the different factor between the groups is significantly more relevant than the variation within the groups, then the factor is assumed to be statistically relevant in explaining the differences



13

The test statistic

• The test statistic is computed as:

F

 s

2

B

2

 s

W

Variance between groups

Variance within groups

• This test statistic compares the weight of the variance explained by the factors to the weight of the variance not explained by the factors



14

Distribution of the

F-statistic (one-tailed test)

Rejection area



15

Characteristics of the

F-distribution

F df df

1 2

)

• Its shape (critical value) changes according to the degrees of freedom (numbers of observation/ groups)

• It is a non-negative statistic (one-tailed test)

• For ANOVA testing: df

1 df

2

=g-1

=n-g

16



Factor



ANOVA in SPSS

Target variable

17

SPSS output

ANOVA

EFS: Total Alcoholic Beverages, Tobacco

Between Groups

Within Groups

Total

Sum of

Squares

6171.784

151535.3

157707.1

df

5

494

499

Variance between

Mean Square

1234.357

306.752

F

4.024

Sig.

.001

Variation decomposition

Degrees of freedom

Variance within p-value < 0.05

The null is rejected



18

Contrasts

• Allows to test hypotheses on specific sub-sets of the treatments (factor combinations).

• They open the way to further explore the sources of variability when the null hypothesis of mean equality is rejected.

• Comparisons are usually based on a theory and planned before the analysis, thus they are also called planned comparisons

.



19

Linear contrasts

• Linear contrasts are linear combinations of the means, allowing one to test other hypotheses than mean equality

• For example, one may want to test whether the mean for group one is double the mean for groups two and three, while the means of groups two and three are equal



20

Linear contrasts

• Contrasts are also useful after rejection of the null hypothesis of mean equality

• Rejection of the null hypothesis means that

at least two means differ, but it does not say which ones actually differ

• Planned comparisons through linear contrasts can help



21

Example

• Test whether chicken expenditures increases linearly with household size

• Check whether there are significant differences:

• Between households with one or two components and households with more components

• Considering the following groups

• Households with one component

• Households with two components

• Households with more than two components

• Considering the following comparison

• Households with four, five, six and seven components have equal means



22

Household sizes and means

Descriptives

In a typical w eek how m uch do you s pend on fres h or frozen chicken (Euro)?

2

3

4

5

0

1

6

7

Total

N

1

82

145

93

87

24

10

1

443

95% C onfidence Interval for

Mea n

Mea n Std. Deviation Std. Error Low er Bound Upp er Bound

4.8000

.

.

.

.

4.2470

5.0548

6.3231

6.7334

7.5613

6.2730

6.7500

5.6677

2.82338

3.41626

4.71695

3.87396

7.64258

3.25606

1.02966

.

4.13383

.3 1179

.2 8371

.4 8912

.4 1533

1.56003

.1 9640

.

3.6266

4.4941

5.3517

5.9078

4.3341

3.9438

5.2817

.

4.8673

5.6156

7.2946

7.5591

10.7884

8.6022

6.0537

.

Minim um Maximum

4.80

4.80

.3 7

.0 0

15.00

20.00

.0 0

.0 0

30.00

18.00

.0 0

.0 0

6.75

.0 0

30.00

10.49

6.75

30.00

7 GROUPS



23

Example

• 1 and 2 components versus 3, 4, 5, 6 and 7

• Weights (they need to sum to 0)

1 =

2 =

3 =

4 =

5 =

6 =

7 =

1

1

1

1

1

-2.5

-2.5



24

Planned comparisons

• Helmert contrasts: the first treatment is compared with all of the remaining treatments, the second treatment will all the remaining treatments but the first, the third treatment will all of the remaining ones but the first two, and so on.

• By looking at the results of this battery of tests, it becomes possible to identify those groups whose difference from the others is most relevant.

• polynomial contrasts: it is possible to tested whether the trend in means follows a linear, quadratic or cubic sequence or any polynomial relationship between the treatments,

• repeated contrasts: each treatment is compared with the one which follows

• reverse Helmert contrasts (or difference contrasts): Helmert contrasts going backwards

• simple contrasts where the user can choose the benchmark treatment between the first and the last category .



25

Post-hoc comparisons

• Linear contrasts are carried out independently from each other

• Post-hoc tests consist in a set of paired comparisons , where the critical values are corrected to account for the problem of inflating the Type I Error risk (rejecting the null hypothesis when it is true) measured by the cumulative

Type I error or familiwise error.

• The approach to correcting the critical values determines the Type of test being used. In SPSS:

– Scheffe’s test

– Bonferroni’s test

– Tukey’s test.

26



Some post-hoc tests

• Scheffe test : simultaneous comparisons for all potential pair-wise and linear combinations of treatments using F critical values corrected to account for the number of treatment being compared

• Bonferroni post-hoc method: (1) run the usual pair-wise ttests; (2) to account for the inflated Type one error rate an adjustment is provided by dividing the family-wise error by the number of tests.

• Tukey’s test: also known as an Honestly Significant

Difference or HSD test, it can be used when samples are of equal size, but statistical packages usually provide variants for unequal sizes. With this test, significant differences are identified through an adjusted Studentized range

distribution (an extension of the Student t statistic which uses pooled estimate of the standard errors)

• More tests on the textbook



27

Effect size and power

• The experimental factor matters, but how much?

( effect size )

• Larger F statistics do not necessarily imply larger effect sizes – because they also depend on sample sizes

• A typical measure of effect size is h 2 (the ratio between variation between and total variation)

• The power of a test is 1b where b is the probability of non-rejecting the null hypothesis when the alternative is true (Type II error)



28

Which post-hoc test?

• One should check the probabilities of Type I error and power (Type II errors)

• There is a trade-off between power and

Type I error

• Conservative tests: low Type I error, low power

(Scheffé, Bonferroni)

• Tukeys test more appropriate for a large number of means



29

ANOVA in SPSS

Target variable (scale)



Planned comparisons

Factor (categorical)

Post-hoc tests

30

Planned comparisons (contrasts)

w

1



L

 w

2



ML

 w

3



MH

 w

4



H



0

The polynomial contrast assumes that the mean follows a given polynomial (linear, quadratic, etc.)

Note: the null hypothesis is that the polynomial contrast does not hold

Other contrasts

Insert a weight for each subgroup

Note: the null hypothesis is that the contrast holds…

Click here to insert other sets of weight (one set of weight per comparison)



31

ANOVA output

Descriptives

EFS: Total Alcoholic Beverages, Tobacco

Low income

Medium-low income

Medium-high income

High income

Total

N

125

125

125

125

500

Mean

7.467

11.381

13.040

18.789

12.669

95% Confidence Interval for

Mean

Std. Deviation Std. Error Lower Bound Upper Bound

12.8693

1.1511

5.188

9.745

17.9038

16.9137

1.6014

1.5128

8.212

10.046

14.551

16.035

20.8025

17.7777

1.8606

.7950

15.106

11.107

22.472

14.231

Minimum Maximum

.0

70.0

.0

.0

93.9

79.4

.0

.0

92.5

93.9

EFS: Total Alcoholic Beverages , Tobacco

Between

Groups

Within Groups

Total

(Combined)

Linear Term Contras t

Deviation

Sum of

Squares

8289.482

7932.717

356.765

149417.6

157707.1

ANOVA df

496

499

3

1

2

Mean Square

2763.161

7932.717

178.382

301.245

F

9.172

26.333

.592

Sig.

.000

.000

.554

Mean equality is rejected

The means are compatible with a linear polynomial

And not compatible with a non-linear one



32

Contrast

1

2

Output

Contrast Coefficients

Anonymised hhold inc + allowances (Banded)

Low income

0

1

Medium-low income

1

0

Medium-high income

-1

0

High income

0

-1

EFS: Total Alcoholic

Beverages, Tobacco

As sume equal variances

Does not as sume equal variances

2

1

Contrast

1

2

Contrast Tests

Value of

Contrast Std. Error

-1.659

2.1954

-11.322

2.1954

-1.659

2.2029

-11.322

2.1879

t

-.756

-5.157

-.753

-5.175

df

496

496

247.202

206.788

Sig. (2-tailed)

.450

.000

.452

.000

The first contrast (0, 1, -1, 0) holds (not rejected)

The second contrast (1, 0, 0, -1) (rejected)



33

SPSS Post-hoc tests



34

SPSS output (post-hoc tests)

Multiple Comparisons

Dependent Variable: EFS: Total Alcoholic Beverages , Tobacco

Tukey HSD

Bonferroni

(I) Anonymised hhold inc

+ allowances (Banded)

Low income

Medium -low income

Medium -high incom e

High incom e

Low income

Medium -low income


High incom e

(J) Anonym ised hhold inc

+ allowances (Banded)

Medium -low income


High incom e

Low income


High incom e

Low income

Medium -low income

High incom e

Low income

Medium -low income


Medium -low income


High incom e

Low income


High incom e

Low income

Medium -low income

High incom e

Low income

Medium -low income


*. The mean difference is s ignificant at the .05 level.

Std. Error

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

2.1954

Mean

Difference

(I-J)

-3.9147

-5.5737

-11.3224*

3.9147

-1.6590

-7.4077*

5.5737

1.6590

-5.7487*

11.3224*

7.4077*

5.7487*

-3.9147

-5.5737

-11.3224*

3.9147

-1.6590

-7.4077*

5.5737

1.6590

-5.7487

11.3224*

7.4077*

5.7487

Sig.

.283

.055

.000

.283

.874

.004

.045

.451

.069

.000

.004

.055

.874

.045

.000

.451

1.000

.005

.069

1.000

.055

.000

.005

.055

95% Confidence Interval

Lower Bound Upper Bound

-9.574

1.745

-11.233

.086

-16.982

-1.745

-7.318

-5.663

9.574

4.000

-13.067

-.086

-4.000

-11.408

5.663

1.748

.089

-9.730

-11.389

-17.138

-1.748

11.233

7.318

-.089

16.982

13.067

11.408

1.901

.242

-5.507

-1.901

-7.474

-13.223

-.242

-4.156

-11.564

5.507

1.592

-.067

9.730

4.156

-1.592

11.389

7.474

.067

17.138

13.223

11.564

Results for each paired comparison are reported and significance level adjusted



35

ANOVA: Fixed versus random effects

1. Explore differences in monthly food expenditure for different geographical regions

2. Explore differences in monthly food expenditure according to the point of purchase for the last food shopping

• 1. is a fixed effect which implies that the researcher can fully control the factor (treatment)

• 2. is a random effect where the factor (treatment) cannot be fully controlled and is subject to a

(random) measurement error



36

ANOVA assumptions

Two key assumptions are needed for running analysis of variance without risks

1) that the sub-samples defined by the treatment are independent

2) that no big discrepancies exist in the variances of the different sub-samples

• Normality within the sub-sample: within limits, departure from normality is not a serious issue

• Different variances: results are still reliable if the sizes of sub-samples are equal

• Both variances and sample sizes differ: high risk of biased results

• Adjustments: Brown-Forsythe test and/or the Welch test instead of the usual F test



37

Adjustments for violated assumptions in

SPSS

Click on OPTIONS to request descriptive stats for a random effect model , Brown-Forsythe and Welch tests (plus more plots and descriptive statistics)



38

Non-parametric ANOVA tests

• Exclusive samples extracted from the same population

• Kruskal–Wallis test: extends the Mann-Whitney test to the case of a higher number of sub-samples. It tests the null hypothesis that all the sub-populations have the same distribution function.

• Jonckheere-Terpstra test: the same null hypothesis, but against the alternative that an increase in treatment leads to an increase in the (median of the) dependent variable.

• Related samples (the same respondent may appear in several treatment sub-samples)

• Friedman test, Kendall test or Cochran Q test, extend to the multiple sample case some of the non-parametric tests for mean comparisons



39

Non-parametric ANOVA in SPSS-

Exhaustive sub-samples Related samples



40

Number of target variables

1

1

2 or more

1

2 or more

1

1

1

1

1

The class of ANOVA techniques

Number of factors

1

2 or more

1 or more

2 or more

2 or more

1 or more

1 or more

1 or more

1

1

Measurement of factors Technique nominal / ordinal independent samples nominal / ordinal independent samples nominal / ordinal independent samples nominal / ordinal and continuous, independent samples nominal / ordinal and continuous, independent samples nominal / ordinal repeated samples nominal / ordinal mixed samples

Nominal / ordinal random effects nominal / ordinal independent samples, nonnormal data and/or non-homogeneous independent samples nominal / ordinal independent samples, nonnormal data and/or non-homogeneous related samples

One-way ANOVA

Factorial ANOVA

MANOVA

ANCOVA

MANCOVA

Repeated ANOVA

Mixed ANOVA

Variance Component Model

Non-parametric tests: Kruskal–Wallis test or

Jonckheree-Terpstra test

Non-parametric tests: Friedman, Cochran Q or Kendall's test



41

Other ANOVA designs

• Multi-way (factorial) ANOVA

• Multivariate ANOVA (MANOVA)

• (Multivariate) Analysis of Covariance

(MANCOVA)



42

General linear model

• One-way ANOVA is equivalent to a linear model, where the target variable is the dependent variable and then each of the treatments is transformed into a dummy

variable which assumes a value of one if respondents are subject to that treatment.

This means that they belong to that economic condition and are zero otherwise.



43

GLM example

Target variable: Alcohol and tobacco expenditure y i

Factor: employment status

 b

0

 b

1

SE i

 b

2

FT i

 b

3

PT i

 b

4

UN i

 b

5

RE i

 b

6

UA i

  i

• y i is the amount spent in alcohol and tobacco by the i-th respondent

• SE i

=1 if the respondent is self-employed

• FT i

• PT i

=1 for full-time employees

=1 for part-time employees

• UN i

• RE i

=1 for unemployed resepondents

=1 for retired or inactive respondents and

• UA i

=1 for those under working age



44

Tests on the GLM coefficients

• T-test on each coefficient: bivariate mean comparison

• F-test: one-way ANOVA

Other analyses of variance

Multi-way (Factorial) ANOVA: More than one factor

(interactions)

MANOVA: More than one target variable: allows one to test whether the factors lead to significant differences in a set of variables.



45

ANCOVA

• A final generalization which is quite interesting for consumer research is the

Analysis of Covariance (ANCOVA), which is the appropriate technique when some of the factors are continuous quantitative variables instead of being measured on a nominal or ordinal scale



46

GLM and ANOVA techniques in SPSS

Univariate GLM: ANOVA, n-way

ANOVA, ANCOVA

Multivariate GLM: MANOVA,

MANCOVA



47



Univariate

GLM

Target variable

Factors (more than one for n-way ANOVA, random factors are allowed)

Scale variables for

ANCOVA

48

Multivariate GLM

More than one target variable for MANOVA or MANCOVA



49

Statistics for Marketing and Consumer Research

Analysis of variance

Tests on multiple hypotheses

Analysis of Variance

Example

Analysis of Variance

One-way ANOVA

Null and alternative hypothesis for

ANOVA

Arranging data for ANOVA

The statistical distribution to carry out

ANOVA

One-way ANOVA: data

Measuring and decomposing the total variation

Variance decomposition

The basic principle of the ANOVA

The test statistic

Distribution of the

F-statistic (one-tailed test)

Characteristics of the

F-distribution

F df df

ANOVA in SPSS

SPSS output

Contrasts

Linear contrasts

Linear contrasts

Example

Household sizes and means

Example

Planned comparisons

Post-hoc comparisons

Some post-hoc tests

Effect size and power

Which post-hoc test?

ANOVA in SPSS

Planned comparisons (contrasts)

ANOVA output

Output

SPSS Post-hoc tests

SPSS output (post-hoc tests)

ANOVA: Fixed versus random effects

ANOVA assumptions

Adjustments for violated assumptions in

SPSS

Non-parametric ANOVA tests

Non-parametric ANOVA in SPSS-

The class of ANOVA techniques

Other ANOVA designs

General linear model

GLM example

Tests on the GLM coefficients

ANCOVA

GLM and ANOVA techniques in SPSS

Univariate

GLM

Multivariate GLM

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