Analysis of Variance Bidin Yatim, PhD Exeter, MSc Aston, BSc Nottingham Statistics Department Facult

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Analysis of VarianceBidin Yatim, PhD (Exeter),
MSc (Aston), BSc (Nottingham)Statistics
DepartmentFaculty of Quantitative Sciences

• Data Analysis Using SPSS
• Topic 6

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Objectives

• Perform and interpret a one-factor independent
  measures ANOVA
• Understand the necessary assumptions for a
  one-factor independent measures ANOVA
• Perform and interpret post-hoc test for a
  one-factor independent measures ANOVA
• Perform and interpret a one-factor repeated
  measures ANOVA
• Understand the necessary assumptions for a
  one-factor repeated measures ANOVA

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Research Problem

• Does the temperature of the lecture hall affect
  the rate at which students fall asleep during
  class?
• IV or Factor is Room Temperature
• Three Levels 50, 70, 90 degrees
• DV is Reaction Time
• How many minutes after the start of lecture does
  the first student fall asleep?

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Raw Data
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Introduction

• Analysis of variance compares two or more
  populations of interval data.
• Specifically, we are interested in determining
  whether differences exist between the population
  means.
• The procedure works by analyzing the sample
  variance.

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One Way Analysis of Variance

• The analysis of variance is a procedure that
  tests to determine whether differences exits
  between two or more population means.
• To do this, the technique analyzes the sample
  variances

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One Way Analysis of Variance Example 1

• An apple juice manufacturer is planning to
  develop a new product -a liquid concentrate.
• The marketing manager has to decide how to market
  the new product.
• Three strategies are considered
• Emphasize convenience of using the product.
• Emphasize the quality of the product.
• Emphasize the products low price.

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One Way Analysis of Variance Example 1

• An experiment was conducted as follows
• In three cities an advertisement campaign was
  launched .
• In each city only one of the three
  characteristics (convenience, quality, and price)
  was emphasized.
• Weekly sales were recorded for 20 weeks following
  the beginning of the campaigns.

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One Way Analysis of Variance
Weekly sales
Weekly sales
Weekly sales
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One Way Analysis of Variance

• Solution
• The data are interval
• The problem objective is to compare mean sales in
  three cities.
• We hypothesize that the three population means
  are equal

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Defining the Hypotheses

• Solution

H0 m1 m2 m3 H1 At least two means differ
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Notation
Independent samples are drawn from k populations
(treatments).
X11 x21 . . . Xn1,1
X12 x22 . . . Xn2,2
X1k x2k . . . Xnk,k
Sample size
Sample mean
X is the response variable. The variables
value are called responses.
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Terminology

• In the context of this problem
• Response variable weekly salesResponses
  actual sale valuesExperimental unit weeks in
  the three cities when we record sales
  figures.Factor the criterion by which we
  classify the populations (the treatments). In
  this problems the factor is the marketing
  strategy.
• Factor levels the population (treatment)
  names. In this problem factor levels are the
  three marketing strategies.

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The rationale of the name of Analysis of Variance
(ANOVA)

• We are testing the different between means but
  why ANOVA?
• Two types of variability are employed when
  testing for the equality of the population means
  Within Samples and Between Samples

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One Way Analysis of Variance
Graphical demonstration Employing two types of
variability Within Samples and Between Samples
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One Way Analysis of Variance
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16 15 14
11 10 9
The sample means are the same as before, but the
larger within-sample variability makes it harder
to draw a conclusion about the population means.
A small variability within the samples makes it
easier to draw a conclusion about the population
means.
Treatment 1
Treatment 2
Treatment 3
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Rationale 1 Variability Between Sample

• If the null hypothesis is true, we would expect
  all the sample means to be close to one another
  (and as a result, close to the grand mean).
• If the alternative hypothesis is true, at least
  some of the sample means would differ.
• Thus, we measure variability between sample means
  (and hence MSTr).

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Rationale II Variability Within

• Large variability within the samples weakens the
  ability of the sample means to represent their
  corresponding population means.
• Therefore, even though sample means may markedly
  differ from one another, we have to consider the
  within samples variability (and hence MSE).

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Test Statistics (F), Critical Value Rejection
Criterion
And finally
the hypothesis test
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The F test
Ho m1 m2 m3 H1 At least two means differ
Test statistic F MSTr/ MSE
3.23
Since 3.23 gt 3.15, there is sufficient evidence
to reject Ho in favor of H1, and argue that at
least one of the mean sales is different than
the others.
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The F test p- value

• Use SPSS to find the p-value
• fx Statistical Table
  DIST(3.23,2,57) .0467

p Value P(Fgt3.23) .0467
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Hey! Lets get our hand dirty Using SPSS.
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One Way Analysis of Variance Using SPSS

• Suppose we want to know whether students who have
  to work many hours outside school to support
  themselves find their grade suffering.
• We examine this question by comparing the GPAs of
  students who work various hours outside school.
• Lets examine this question using data in our
  Student file. FilegtOpengt Student

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One Way Analysis of Variance Using SPSS

• First examine the average GPA for each of the
  three work categories (0 hrs,1-19hrs,gt20hrs)-WorkC
  at
• GraphgtBoxplot then choose Simple and click
  Define. Select GPA as the variable and WorkCat
  for the Category Axis. Click
• Option

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After Clicking Options, click off Display
groups defined by missing value, and click
Continue then OK.

• Youll get this

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What is the Box-plot telling us?

• Some variation across the groups
• See median GPAs (dark line in the middle of box)
  differ slightly between groups. Thats natural.
  Probably because of sampling error.
• So, should we attribute the observed difference
  to sampling error or they genuinely differ?
• Neither box-plot nor the median offer decisive
  evidence. Hence we need ANOVA.

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One Way Analysis of Variance Using SPSS

• We are testing
• H1 At least two means differ
• Before attempting ANOVA, need to review the ANOVA
  assumptions. (i) Independent samples (ii)
  Normality (iii) Variances equality. We can test
  both (ii) (iii).
• AnalyzegtDescriptive StatisticsgtExplore

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AnalyzegtDescriptive StatisticsgtExplore

• In the Explore dialog box, select GPAs as the
  dependent List variable, WorkCat as the Factor
  List variable and Plot as the Display. Next,
  click Plot
• We are interested in a
• normality test, select
• Deselect this
• select this only. Click Continue
  and OK. See next slide

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The Output has several parts, let focus on the
tests of normality

• The Kolmogorov-Smirnov test assesses whether
  there is significant departure from normality in
  the population distribution of the 3 groups. Null
  hypothesis Distributions are normal.
• Look at the p-values, all are gt0.05. Hence no
  evidence of non-normality.

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One Way Analysis of Variance Using SPSS

• We still need to validate the homogeneity of
  variance assumption. We do this within ANOVA.
• AnalyzegtCompare MeansgtOne-Way ANOVA
• Dependent List variable
• is GPA and Factor
• variable is WorkCat

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One Way Analysis of Variance Using SPSS

• Click Option under Statistics, select
  Descriptive and Homogeneity of variance test.
  Click Continue OK
• One-Way ANOVA output
• Consists many parts.
• Focus on hence do not
• reject Ho Variances are equal.

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Normality Homogeneity of variances assumptions
met hence

• Let find out whether students who work various
  hours outside school differ in their GPAs.
• The P-value of .000 is very small, hence we
  reject Ho and conclude that
• the means GPAs are not all the same. Where are
  the differences? Hence Post-Hoc test

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One Way Analysis of Variance Using SPSS Post-hoc
Test

• Before doing Post-hoc test, lets look at the
  group means, please comment.
• Eyeballing group means cannot tell us decisively
  if significant differences exist.
• Many options exist. Commonly used- Tukeys It
  tests compares all pairs of group means without
  increasing the probability of Type 1 Error.
• AnalyzegtCompare MeansgtOne-Way ANOVA

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One Way Analysis of Variance Using SPSS Post-hoc
Test

• The variables are still selected. Click
  Post-Hoc, select only Tukey then Continue and OK

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One Way Analysis of Variance Using SPSS Post-hoc
Test

• The results of Tukey Test could be summarized as
  follows
• Group 1-19hrs had better GPAs than 0hrs
• Group 1-19hrs is comparable to 20hrs
• Group 20hrs is comparable to 0hrs.

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One Way Analysis of Variance Using SPSS Exercise

• FilegtOpengtGSS94
• This is the extract from the 1994 General Social
  Survey.
• One variable in the file groups respondent into
  four age categories.Do the mean number of tv
  hours vary by age group?

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Analysis of Variance Experimental Designs

• Several elements may distinguish between one
  experimental design and others.
• Either independent or dependent samples used.
• The number of factors.
• Each characteristic investigated is called a
  factor.
• Each factor has several levels.

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One-Factor Repeated Measures Analysis of Variance

• There are many situations where we are interested
  in examining the same sample across three or more
  treatments (several measurements on the same set
  of individuals).
• Do blood pressure changes during various stressor
  tasks?
• Lets examine using data in file called BP.

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One-Factor Repeated Measures Analysis of Variance

• We are testing
• H1 At least two means differ
• Repeated Measures ANOVA requires 4 conditions
  (i) Independent observations within each
  treatment (ii) normality (iii) homogeneous
  variances (iv) sphericity

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One-Factor Repeated Measures Analysis of
Variance BP file

• BP file contains
• Dbprest diastolic BP at rest
• Dbpma diastolic BP during mental arithmetic
• Dbpcp diastolic BP while immersing a hand in ice
  water.
• Test the normality of the variables
• See next slides for sphericity test.

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One-Factor Repeated Measures Analysis of Variance
Using SPSS

• AnalyzegtGeneral Linear Modelgt Repeated Measures
  These instructions will prompt the dialog box
  shown. Assign our repeated measure (also called
  within subject factor) and indicate
• the number of levels as 3.
• Factor1(3) will appear here
• after we click Add. After
• clicking Add, click Define
• refer next slide watch a
• new dialog box.

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One-Factor Repeated Measures Analysis of Variance
Using SPSS

• Select a specific level for our repeated measure
  dbprest, dbpma, and dbpcp in the order indicated.
• Click on Option and
• select Descriptive
• Statistics, Continue
• OK

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One-Factor Repeated Measures Analysis of Variance
Using SPSS

• The output has several parts First look at
  Mauchlys sphericity test. Itll determine which
  ANOVA test to use. Ho is that the correlations
  among the 3 measures are equal. Look at p-value,
  hence reject Ho.
• Sphericity assumption is not met.

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One-Factor Repeated Measures Analysis of Variance
Using SPSS

• Then look at output labeled Test of Within
  Subjects Effects, 1st line of factor1 reads
  Sphericity Assumed. This is the F test line to
  refer if sphericity condition met.
• So? Many other F tests to choose from The
  Greengouse-Geisser adjusted test is commonly
• used. Look
• at p-value.
• So what?

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One-Factor Repeated Measures Analysis of Variance
Using SPSS CONCLUSION

• The means for Diastolic BP (DBP) for the 3 tasks
  are not the same. Hence it changes significantly
  during the various mental and physical stressors
  investigated in this study.
• But where the differences are? Determine
  manually..

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One-Factor Repeated Measures Analysis of
Variance Exercise

• Does systolic BP change significantly during the
  three tasks examined in this study? The 3 tasks
  are at rest (sbprest), performing mental
  arithmetic (sbpma), and immersing a hand in ice
  water (sbpcp).

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Two Way Analysis of Variance

• Objectives
• Perform and interpret a two-factor independent
  measures ANOVA
• Understand the necessary assumptions for a
  two-factor independent measures ANOVA
• Understand and interpret statistical main effects
• Understand and interpret statistical interactions

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Two Way Analysis of Variance

• Previously we learned how to perform ANOVA for
  research situations involving a single IV.
• However, in many situations we want to consider
  two IV simultaneously. The analysis used is a
  two-way analysis of variance.

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One - way ANOVA Single factor
Two - way ANOVA Two factors
Response
Response
Treatment 3 (level 1)
Treatment 2 (level 2)
Treatment 1 (level 3)
Level 3
Level2
Factor A
Level 1
Level 1
Level2
Factor B
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Two Way Analysis of Variance Research Problem

• Prior research found that people at risk of
  having hypertension (HBP) showed changes in
  cardiovascular responses to various stressors.
• A researcher wants to explore this finding
  further by looking at several variables that
  might be implicated, in particular, a persons
  sex and whether a person has a parent with HBP.
• Various DV can be measured. We focus on systolic
  BP during mental arithmetic task.

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Two Way Analysis of Variance Hypothesis

• Two-way ANOVA will test for
• Mean difference between levels of 1st factor
  (here, comparing systolic BP during mental
  arithmetic (sbpma) between gender)
• Mean difference between levels of 2nd factor
  (here, comparing systolic BP during mental
  arithmetic (sbpma) for individuals having parent
  with HBP or not)
• Any other mean differences as a result of a
  unique combination of the two factors called
  interaction effects.

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Two Way Analysis of Variance Hypothesis

• 1st two hypothesis tests called tests for the
  main effects. Ho there are no differences
  between the levels of the factor.
• 3rd hypothesis test for the interaction between
  the two factors. Ho there is no interaction
  between the factors.
• The three tests are independent. The outcome of
  one does not effect the other.

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Two Way Analysis of Variance Using SPSS,
FilegtOpengtBP

• AnalyzegtGeneral Linear ModelgtUnivariate
• Select systolic bp mental arithmetic sbpma as
  the Dependent Variable and sex and parental
• hypertension PH
• as Fixed Factors.
• Click Option and..

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Two Way Analysis of Variance Using SPSS

• Under Display, select Descriptive Statistics and
  homogeneity tests. Click Continue then OK

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Two Way Analysis of Variance Using SPSS

• Univariate ANOVA output has several parts
  (descriptive statistics, Levenes test of
  equality of variances, and tests of
  between-subjects effects).
• 1st look at Levenes it tests variance
  homogeneity. Its
• P-value is .225 gt .05,
• hence the data do not
• violate variance homo-
• geneity assumption.
• So, can proceed to interpret ANOVA.

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Two Way Analysis of Variance Using SPSS

• Lets check if sbpma is related to a persons
  gender, parental history of ph or some other
  combination of these factor. P-value for testing
  1st, 2nd and 3rd hypotheses are shown in red,
  blue and green line
• respectively.

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Two Way Analysis of Variance Using SPSS. A
conclusions

• Reject the 1st hypothesis, hence there is a
  significant main effect for gender.
• Reject 2nd hypothesis, hence there is a
  significant main effect for parental history.
• P-value for testing the interaction effect is
  greater than .05, hence do not reject the Ho and
  conclude that there is no significant interaction
  between the two factors.

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Two Way Analysis of Variance Using SPSS. A
conclusions

• What exactly do these results mean?
• We have two significant main effects and a non-
  significant interaction.
• One very helpful way to make sense of a
  two-factor ANOVA results is to graph the data.
• GraphsgtInteractiongtBarDrag sbpma to the vertical
  axis, PH to the horizontal axis and sex to the
  Panel Variables.
• Click Titles tab to title your bar chart and put
  your name in the caption.

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• Bars are about
• the same hght.
• Hard to differentiate

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Editing the bar chart

• Lets use the interactive graphing capabilities
  to make a bar chart where the y-axis does not
  start with zero, and thus show the differences
  more clearly. The lowest sbpma is 118, so let
  start the y-axis at 118.
• Double click anywhere on your chart. Find Chart
  Manager and click on it.
• In the Chart Manager, click on Scales Axis and
  then on Edit
• In the Scale Axis dialog box, under Scale, find
  Minimum and deselect Auto. Type 118. Click OK

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Edited bar chart.. At last

• So? We can see that, males have higher SBP,
  regardless of parental hypertension.
• Person having a parent with hypertension have
  higher SBP, regardless of gender.
• Both factors effect SBP separately but not their
  combination.

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EXERCISE

• FilegtOpengtBP
• Is heart rate while immersing a hand in ice water
  hrcp related to a persons sex, parental
  hypertension PH, or some combination of these
  factors?

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End of Part SixSee U LaterPlease Dont Go
Away What if the data is not normal?