RESEARCH STATISTICS Jobayer Hossain Larry Holmes, Jr, November 20, 2008 Analysis of Variance.

RESEARCH STATISTICSRESEARCH STATISTICS

Jobayer Hossain

Larry Holmes, Jr,

November 20, 2008

Analysis of Variance

Experimental Design TerminologyExperimental Design Terminology

An Experimental Unit is the entity on which measurement or an

observation is made. For example, subjects are experimental units

in most clinical studies.

Homogeneous Experimental Units: Units that are as uniform as

possible on all characteristics that could affect the response.

A Block is a group of homogeneous experimental units. For

example, if an investigator had reason to believe that age might be a

significant factor in the effect of a given medication, he might

choose to first divide the experimental subjects into age groups,

such as under 30 years old, 30-60 years old, and over 60 years old.


A Factor is a controllable independent variable that is being

investigated to determine its effect on a response. E.g.

treatment group is a factor.

Factors can be fixed or random

– Fixed -- the factor can take on a discrete number of values

and these are the only values of interest.

– Random -- the factor can take on a wide range of values

and one wants to generalize from specific values to all

possible values.

Each specific value of a factor is called a level.


A covariate is an independent variable not manipulated by the

experimenter but still affecting the response. E.g. in many

clinical experiments, the demographic variables such as race,

gender, age may influence the response variable significantly

even though these are not the variables of interest of the study.

These variables are termed as covariate.

Effect is the change in the average response between two

factor levels. That is, factor effect = average response at one

level – average response at a second level.


Interaction is the joint factor effects in which the effect of one

factor depends on the levels of the other factors.

No interaction effect of factor A and B

Interaction effect of factor A and B


Randomization is the process of assigning

experimental units randomly to different experimental

groups.

It is the most reliable method of creating

homogeneous treatment groups, without involving

potential biases or judgments.

Experimental Design Experimental Design TerminologyTerminology

A Replication is the repetition of an entire experiment or portion of

an experiment under two or more sets of conditions.– Although randomization helps to insure that treatment groups are as similar as

possible, the results of a single experiment, applied to a small number of experimental units, will not impress or convince anyone of the effectiveness of the treatment.

– To establish the significance of an experimental result, replication, the repetition of an experiment on a large group of subjects, is required.

– If a treatment is truly effective, the average effect of replicated experimental units will reflect it.

– If it is not effective, then the few members of the experimental units who may have reacted to the treatment will be negated by the large numbers of subjects who were unaffected by it.

– Replication reduces variability in experimental results and increases the significance and the confidence level with which a researcher can draw conclusions about an experimental factor.


A Design (layout) of the experiment includes the choice of factors and factor-

levels, number of replications, blocking, randomization, and the assignment of

factor –level combination to experimental units.

Sum of Squares (SS): Let x1, …, xn be n observations. The sum of squares of

these n observations can be written as x12 + x2

2 +…. xn2. In notations, ∑xi

2. In a

corrected form this sum of squares can be written as (xi -x )2.

Degrees of freedom (df): Number of quantities of the form – Number of

restrictions. For example, in the following SS, we need n quantities of the form

(xi -x ). There is one constraint (xi -x ) = 0. So the df for this SS is n – 1.

Mean Sum of Squares (MSS): The SS divided by it’s df.

Experimental Design TerminologyExperimental Design Terminology The analysis of variance (ANOVA) is a technique of decomposing the

total variability of a response variable into:

– Variability due to the experimental factor(s) and…

– Variability due to error (i.e., factors that are not accounted for in the experimental

design).

The basic purpose of ANOVA is to test the equality of several means.

A fixed effect model includes only fixed factors in the model.

A random effect model includes only random factors in the model.

A mixed effect model includes both fixed and random factors in the

model.

The basic ANOVA situationThe basic ANOVA situation

Two type of variables: Quantitative responseCategorical predictors (factors),

Main Question: Do the (means of) the quantitative variable depend on which group (given by categorical variable) the individual is in?

If there is only one categorical variable with only 2 levels (groups):

• 2-sample t-test

ANOVA allows for 3 or more groups

One-way analysis of VarianceOne-way analysis of Variance

One factor of k levels or groups. E.g., 3 treatment groups in a drug study.

The main objective is to examine the equality of means of different groups.

Total variation of observations (SST) can be split in two components:

variation between groups (SSG) and variation within groups (SSE).

Variation between groups is due to the difference in different groups. E.g.

different treatment groups or different doses of the same treatment.

Variation within groups is the inherent variation among the observations

within each group.

Completely randomized design (CRD) is an example of one-way analysis of

variance.

One-way analysis of varianceOne-way analysis of variance

Consider a layout of a study with 16 subjects that intended to compare 4 treatment groups (G1-G4). Each group contains four subjects.

S1 S2 S3 S4

G1 Y11 Y12 Y13 Y14

G2 Y21 Y22 Y23 Y24

G3 Y31 Y32 Y33 Y34

G4 Y41 Y42 Y43 Y44

One-way Analysis: Informal InvestigationOne-way Analysis: Informal Investigation

Graphical investigation: • side-by-side box plots• multiple histograms

Whether the differences between the groups are significant depends on

• the difference in the means• the standard deviations of each group• the sample sizes

ANOVA determines P-value from the F statistic

One-way Analysis: Side by Side One-way Analysis: Side by Side BoxplotsBoxplots


Model:

Assumptions:

– Observations yij are independent.

– eij are normally distributed with mean zero and constant standard deviation.

– The second (above) assumption implies that response variable for each group is normal (Check using normal quantile plot or histogram or any test for normality) and standard deviations for all group are equal (rule of thumb: ratio of largest to smallest are approximately 2:1).

error. theis andmean general theis

group, ofeffect theis

group, ofn observatio theis where,

ij

i

ij

ijiij

e

ith

jthithy

ey


Hypothesis:Ho: Means of all groups are equal.

Ha: At least one of them is not equal to other.

– doesn’t say how or which ones differ.

– Can follow up with “multiple comparisons”

Analysis of variance (ANOVA) Table for one way classified data

Sources of Variation

Sum of Squares

df Mean Sum of Squares

F-Ratio

Group SSG k-1 MSG=SSG/k-1 F=MSG/MSE

Error SSE n-k MSE=SSE/n-k

Total SST n-1

A large F is evidence against H0, since it indicates that there is more difference between groups than within groups.

Pooled estimate for variancePooled estimate for variance

sp2

(n1 1)s12 (n2 1)s2

2 ... (nI 1)sI2

n I

sp2

(df1)s12 (df2)s2

2 ... (df I )sI2

df1 df2 ... df I

The pooled variance of all groups can be estimated as the weighted average of the variance of each group:

MSEDFE

SSEsp 2

so MSE is the pooled estimate of variance

Multiple comparisonsMultiple comparisons

If the F test is significant in ANOVA table, then we intend

to find the pairs of groups are significantly different.

Following are the commonly used procedures:– Fisher’s Least Significant Difference (LSD)

– Tukey’s HSD method

– Bonferroni’s method

– Scheffe’s method

– Dunn’s multiple-comparison procedure

– Dunnett’s Procedure

One-way ANOVA - DemoOne-way ANOVA - Demo

MS Excel:

– Put response data (hgt) for each groups (grp) in side by side

columns (see next slides)

– Select Tools/Data Analysis and select Anova: Single Factor

from the Analysis Tools list. Click OK.

– Select Input Range (for our example a1: c21), mark on Group by

columns and again mark labels in first row.

– Select output range and then click on ok.

One-way ANOVA MS-Excel Data One-way ANOVA MS-Excel Data LayoutLayout

grp1 grp2 grp352.50647 47.83167 43.8221143.14426 43.69665 40.4135955.91853 40.73333 57.6574845.68187 46.56424 41.9620754.76593 53.03273 49.0805145.27999 56.41127 55.9779741.95513 43.69635 47.7141943.67319 50.92119 41.7591244.01857 40.36097 46.2185944.54295 51.95264 53.61966

46.1765 57.1638 49.6348440.02826 50.98321 57.6822958.09449 49.23148 49.0847143.25757 48.01014 40.5986642.07507 45.98231 38.9905546.80839 55.32514 54.7428643.80479 47.21837 53.7462457.60508 40.30104 44.8250742.47022 50.56327 45.85581

57.4945 55.55145 41.36863

One-way ANOVA MS-Excel output: One-way ANOVA MS-Excel output: height on treatment groupsheight on treatment groups

Anova: Single Factor

SUMMARYGroups Count Sum Average Variance

grp1 20 949.3017 47.46509 36.88689grp2 20 975.5313 48.77656 28.10855grp3 20 954.7549 47.73775 37.50739

ANOVASource of Variation SS df MS F P-value F critBetween Groups 19.15632 2 9.578159 0.280329 0.756571 3.158846Within Groups 1947.554 57 34.16761

Total 1966.71 59

One-way ANOVA - DemoOne-way ANOVA - Demo

SPSS:

– Select Analyze > Compare Means > One –Way ANOVA

– Select variables as Dependent List: response (hgt), and Factor:

Group (grp) and then make selections as follows-click on Post Hoc

and select Multiple comparisons (LSD, Tukey, Bonferroni, or

Scheffe), click options and select Homogeneity of variance test,

click continue and then Ok.

One-way ANOVA SPSS output: height One-way ANOVA SPSS output: height on treatment groupson treatment groups

ANOVA hgt

Sum of Squares df Mean Square F Sig.

Between Groups 19.156 2 9.578 .280 .757 Within Groups 1947.554 57 34.168 Total 1966.710 59

Analysis of variance of factorial Analysis of variance of factorial experiment (Two or more factors)experiment (Two or more factors)

Factorial experiment:

– The effects of the two or more factors including their interactions

are investigated simultaneously.

– For example, consider two factors A and B. Then total variation of

the response will be split into variation for A, variation for B,

variation for their interaction AB, and variation due to error.


Model with two factors (A, B) and their interactions:

Assumptions: The same as in One-way ANOVA.

error theis

B of level andA level ofeffect n interactio theis

Bfactor theof level ofeffect theis

Afactor theof level ofeffect theis

mean general theis

)(

ijk

ij

j

i

ijkijjiijk

e

jthithβ)(

jth

ithα

ey


Null Hypotheses:

Hoa: Means of all groups of the factor A are equal.

Hob: Means of all groups of the factor B are equal.

Hoab:(αβ)ij = 0, i. e. two factors A and B are independent


ANOVA for two factors A and B with their interaction AB.

Kpr-1SSTTotal

MSE=SSE/kp(r-1)

kp(r-1)SSEError

MSAB/MSEMSAB=SSAB/ (k-1)(p-1)

(k-1)(p-1)SSABInteraction Effect AB

MSB/MSEMSB=SSB/p-1P-1SSBMain Effect B

MSA/MSEMSA=SSA/k-1k-1SSAMain Effect A

F-RatioMean Sum of Squares

dfSum of Squares


Kpr-1SSTTotal

MSE=SSE/kp(r-1)

kp(r-1)SSEError

MSAB/MSEMSAB=SSAB/ (k-1)(p-1)

(k-1)(p-1)SSABInteraction Effect AB

MSB/MSEMSB=SSB/p-1P-1SSBMain Effect B

MSA/MSEMSA=SSA/k-1k-1SSAMain Effect A

F-RatioMean Sum of Squares

dfSum of Squares


Two-factor with replication - DemoTwo-factor with replication - Demo

MS Excel:

– Put response data for two factors like in a lay out like in the next

page.

– Select Tools/Data Analysis and select Anova: Two Factor with

replication from the Analysis Tools list. Click OK.

– Select Input Range and input the rows per sample: Number of

replications (excel needs equal replications for every levels).

Replication is 2 for the data in the next page.

– Select output range and then click on ok.

Two-factor ANOVA MS-Excel Data Two-factor ANOVA MS-Excel Data LayoutLayout

grp1 grp2 grp3shade1 52.50647 47.83167 43.82211shade1 43.14426 43.69665 40.41359shade1 55.91853 40.73333 57.65748shade1 45.68187 46.56424 41.96207shade1 54.76593 53.03273 49.08051shade1 45.27999 56.41127 55.97797shade1 41.95513 43.69635 47.71419shade1 43.67319 50.92119 41.75912shade1 44.01857 40.36097 46.21859shade1 44.54295 51.95264 53.61966shade2 46.1765 57.1638 49.63484shade2 40.02826 50.98321 57.68229shade2 58.09449 49.23148 49.08471shade2 43.25757 48.01014 40.59866shade2 42.07507 45.98231 38.99055shade2 46.80839 55.32514 54.74286shade2 43.80479 47.21837 53.74624shade2 57.60508 40.30104 44.82507shade2 42.47022 50.56327 45.85581shade2 57.4945 55.55145 41.36863

Two-factor ANOVA MS-Excel output: height on Two-factor ANOVA MS-Excel output: height on treatment group, shades, and their interactiontreatment group, shades, and their interaction

Anova: Two-Factor With Replication

SUMMARYgrp1 grp2 grp3 Totalshade1

Count 10 10 10 30Sum 471.4869 475.201 478.2253 1424.913Average 47.14869 47.5201 47.82253 47.49711Variance 26.773 29.80231 38.11298 29.46458

shade2

Count 10 10 10 30Sum 477.8149 500.3302 476.5297 1454.675Average 47.78149 50.03302 47.65297 48.48916Variance 50.87686 26.02977 41.05331 37.84396

Total

Count 20 20 20Sum 949.3017 975.5313 954.7549Average 47.46509 48.77656 47.73775Variance 36.88689 28.10855 37.50739

ANOVASource of Variation SS df MS F P-value F crit

Sample 14.76245 1 14.76245 0.416532 0.521406 4.01954Columns 19.15632 2 9.578159 0.270254 0.764212 3.168246Interaction 18.9572 2 9.478602 0.267445 0.766341 3.168246Within 1913.834 54 35.44137

Total 1966.71 59

Two-factor ANOVA - DemoTwo-factor ANOVA - Demo

SPSS:

– Select Analyze > General Linear Model > Univariate

– Make selection of variables e.g. Dependent varaiable: response

(hgt), and Fixed Factor: grp and shades.

– Make other selections as follows-click on Post Hoc and select

Multiple comparisons (LSD, Tukey, Bonferroni, or Scheffe),

click options and select Homogeneity of variance test, click

continue and then Ok.

Two-factor ANOVA SPSS output: height on Two-factor ANOVA SPSS output: height on treatment group, shades, and their interactiontreatment group, shades, and their interaction

Between-Subjects Factors

N 1 20 2 20

grp

3 20 1 30 Shades

2 30

Tests of Between-Subjects Effects Dependent Variable: hgt

Source Type III Sum of Squares df Mean Square F Sig.

Corrected Model 52.876(a) 5 10.575 .298 .912 Intercept 138200.445 1 138200.445 3899.410 .000 grp 19.156 2 9.578 .270 .764 Shades 14.762 1 14.762 .417 .521 grp * Shades 18.957 2 9.479 .267 .766 Error 1913.834 54 35.441 Total 140167.155 60 Corrected Total 1966.710 59

a R Squared = .027 (Adjusted R Squared = -.063)

Repeated MeasuresRepeated Measures

The term repeated measures refers to data sets with multiple

measurements of a response variable on the same experimental unit

or subject.

In repeated measurements designs, we are often concerned with

two types of variability:– Between-subjects - Variability associated with different groups of subjects who

are treated differently (equivalent to between groups effects in one-way ANOVA)

– Within-subjects - Variability associated with measurements made on an

individual subject.


Examples of Repeated Measures designs:

A. Two groups of subjects treated with different drugs for whom responses

are measured at six-hour increments for 24 hours. Here, DRUG

treatment is the between-subjects factor and TIME is the within-subjects

factor.

B. Students in three different statistics classes (taught by different

instructors) are given a test with five problems and scores on each

problem are recorded separately. Here CLASS is a between-subjects

factor and PROBLEM is a within-subjects factor.

C. Volunteer subjects from a linguistics class each listen to 12 sentences

produced by 3 text to speech synthesis systems and rate the naturalness

of each sentence. This is a completely within-subjects design with

factors SENTENCE (1-12) and SYNTHESIZER


When measures are made over time as in example A we want to assess:– how the dependent measure changes over time independent of

treatment (i.e. the main effect of time)– how treatments differ independent of time (i.e., the main effect of

treatment)– how treatment effects differ at different times (i.e. the treatment by

time interaction). Repeated measures require special treatment because:

– Observations made on the same subject are not independent of each other.

– Adjacent observations in time are likely to be more correlated than non-adjacent observations

Res

pons

e

Time



Methods of repeated measures ANOVA

– Univariate - Uses a single outcome measure.

– Multivariate - Uses multiple outcome measures.

– Mixed Model Analysis - One or more factors (other than

subject) are random effects.

We will discuss only univariate approach


Assumptions:

– Subjects are independent.

– The repeated observations for each subject follows a

multivariate normal distribution

– The correlation between any pair of within subjects levels

are equal. This assumption is known as sphericity.


Test for Sphericity:

– Mauchley’s test

Violation of sphericity assumption leads to inflated F statistics and hence inflated

type I error.

Three common corrections for violation of sphericity:

– Greenhouse-Geisser correction

– Huynh-Feldt correction

– Lower Bound correction

All these three methods adjust the degrees of freedom using a correction factor

called Epsilon.

Epsilon lies between 1/k-1 to 1, where k is the number of levels in the within subject

factor.

Repeated Measures - SPSS DemoRepeated Measures - SPSS Demo

Analyze > General Linear model > Repeated Measures

Within-Subject Factor Name: e.g. time, Number of Levels:

number of measures of the factors, e.g. we have two

measurements PLUC.pre and PLUC.post. So, number of level is

2 for our example. > Click on Add

Click on Define and select Within-Subjects Variables (time):

e.g. PLUC.pre(1) and PLUC.pre(2)

Select Between-Subjects Factor(s): e.g. grp

Repeated Measures ANOVA SPSS Repeated Measures ANOVA SPSS output:output:

Within-Subjects Factors Measure: MEASURE_1

time Dependent

Variable 1 PLUC.pre 2 PLUC.post

Between-Subjects Factors

N 1 20 2 20

grp

3 20

Mauchly's Test of Sphericity(b) Measure: MEASURE_1

Epsilon(a)

Within Subjects Effect Mauchly's W Approx. Chi-

Square df Sig. Greenhouse-

Geisser Huynh-Feldt Lower-bound time 1.000 .000 0 . 1.000 1.000 1.000

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b Design: Intercept+grp Within Subjects Design: time

Repeated measures ANOVA SPSS Repeated measures ANOVA SPSS outputoutput

Tests of Within-Subjects Effects Measure: MEASURE_1


Sphericity Assumed 172.952 1 172.952 33.868 .000 Greenhouse-Geisser 172.952 1.000 172.952 33.868 .000 Huynh-Feldt 172.952 1.000 172.952 33.868 .000

time

Lower-bound 172.952 1.000 172.952 33.868 .000 Sphericity Assumed 46.818 2 23.409 4.584 .014 Greenhouse-Geisser 46.818 2.000 23.409 4.584 .014 Huynh-Feldt 46.818 2.000 23.409 4.584 .014

time * grp

Lower-bound 46.818 2.000 23.409 4.584 .014 Sphericity Assumed 291.083 57 5.107 Greenhouse-Geisser 291.083 57.000 5.107 Huynh-Feldt 291.083 57.000 5.107

Error(time)

Lower-bound 291.083 57.000 5.107

Tests of Within-Subjects Contrasts Measure: MEASURE_1

Source time Type III Sum of Squares df Mean Square F Sig.

time Linear 172.952 1 172.952 33.868 .000 time * grp Linear 46.818 2 23.409 4.584 .014 Error(time) Linear 291.083 57 5.107

Tests of Between-Subjects Effects Measure: MEASURE_1 Transformed Variable: Average


Intercept 14371.484 1 14371.484 3858.329 .000 grp 121.640 2 60.820 16.328 .000 Error 212.313 57 3.725

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RESEARCH STATISTICS Jobayer Hossain Larry Holmes, Jr, November 20, 2008 Analysis of Variance.

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