08.Two Way Anova

8/13/2019 08.Two Way Anova

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Two-way ANOVA


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Two-Way ANOVA Data Layout

Xijk

Level iFactor

A

Level jFactor

B

Observation k

in each cell

Factor Factor B

A 1 2 ... b

1 X111 X121 ... X1b1X11n X12n ... X1bn

2 X211 X221 ... X2b1

X21n X22n ... X2bn

: : : : :

a Xa11 Xa21 ... Xab1

Xa1n Xa2n ... Xabni = 1,,a

j = 1,,b

k = 1,,nThere are a X b treatment combinations


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Motivating Example:

Capsule Dissolve Time

Suppose are looking at two capsule types

(C or V) & two digestive fluids (Gastric or Duodenal)

Randomly assign 5

capsules of each type toeach of type of digestive

juice and observe

dissolve time.

Xijk = measured dissolve

time for capsule kin

digestive juice i and

capsule type j.

i = 1 or 2 (i.e. G and D)

j = 1 or 2 (i.e. C and V)

k = 1,2,3,4,5


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Plotting the Results ~ Capsule Effect

seconds42.85dissolvetocapsulesVfor typemean timeseconds05.43dissolvetocapsulesCfor typemean time

2

1

XX

TimeUntilBubbles(second

s)

Capsule Type

There appears to be very little difference between the capsule

types in terms of the time it takes them to dissolve.


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Plotting the Results ~ Fluid Effect

seconds40.2juiceduodenalindissolvetocapsulesformean timeseconds7.45juicegastricindissolvetocapsulesformean time

2

1

XX

Fluid Type

Capsules take 5.5 seconds longer on average to dissolve in

gastric juice compared to duodenal juice.


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Preliminary Conclusion

There is very little difference between the capsule

types in terms of the length it time it takes them to

dissolve.

Capsules take about 5 seconds longer on average to

dissolve in gastric juice than in duodenal juice.

THESE CONCLUSIONS ARE

COMPLETELY WRONG!! WHY ?!?


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Plotting the Results ~ Capsule Effect Separately

seconds44.1juiceduodenalindissolvetocapsulesVformean time

seconds36.3juiceduodenalindissolvetocapsulesCformean timeseconds41.6juicegastricindissolvetocapsulesVformean time

seconds8.49juicegastricindissolvetocapsulesCformean time

22

21

12

11

X

XX

X

Clearly the time to dissolve depends on what capsule is being

used and which juice it is being dissolved in.

Type C capsules dissolve faster in duodenal

juice than do type V capsules where for gastric

juice the opposite is true.


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Interactions

The capsule study is an example of situation wherethere is an interaction between the two factors

being studied in terms of their effect on the

numeric response.

An interaction occurs when the effect of one

factor depends on the level of another factor.Here the effect of capsule depends on the type

of digestive juice used to dissolve it and vise

versa.

Type C capsules dissolvefaster than Type V in

duodenal juice, where

opposite is true when

gastric juice is used todissolve the capsules.


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Interactions can mask main effects

The apparent lack of

a capsule effect is

caused by theinteraction of capsule

type and fluid type.We say the interaction masks

the main effect of capsule

type.


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Types of Interactions and Interpreting

Interaction Plots

Here the meanresponse is the same

for both levels of both

factors.

Here both effects are masked

by the interaction. This type

of interaction is called a

difference in direction of the

effects.


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Interaction Plots

Here the meanresponse differs

depending on the level

of B but not A.

Here the A main effect is

masked by the interaction. The

B main effect is significant,

although cannot be talked about

independently of the level of A.


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Interaction Plots

Here the effect of A is the

same for both levels of B.There is minimal separation

between the two profiles for

the levels of B, thus B is not

significant

Here the A main effect is differs

depending on the level of B. The

B main effect is masked by the

interaction as the means for B1

and B2 are the same.


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Interaction Plots

Here the effect of A is the same

for both levels of B and viseversa. The response differs

across the level of both factors

and both differences suggest

significant A & B effects.

Here the A main effect is

differs depending on the levelof B. Neither the A or B

main effects are masked by

the interaction.

This type of interaction is adifference in magnitude the

effect. The direction of A main

effect is the same for both levels

of B, however the A effect is

larger when B is at the 1st level.


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Types of Interactions

In summary there are types of interactions:

Differences in Direction Differences in Magnitude

Always construct an interaction plot to

visualize the interaction or lack thereof !


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Questions of Interest

Generally, the questions of interest here

(i.e. hypotheses to be tested) concern three

questions regarding the potential effects of

the factors on the response variable.

Question 1: Do the effects that factorsA and

B have on the response variable interact, i.e.is there a significant interactionbetween

factorsA and B ?


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If we conclude there is a significant interaction

then we conclude the effects of both factors

A and B are significant!

When we have an interaction we cannot consider

the effect of either factor independently of the

other, therefore both factors matter.


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If there is not a significant interactioneffect then we can consider the maineffects separately, i.e. we ask the

following:

Question 2: Does factorA alone have asignificant effect?

Question 3: Does factor B alone have asignificant effect?


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Tests of Hypotheses

Just as we had Sums of Squares and Mean Squaresin One-way ANOVA, we have the same in Two-

way ANOVA:

Recall, Mean Squares are measures of variabilityacross the levels of the relevant factor of interest.

In balanced Two-way ANOVA, we measure the

overall variability in the data by:1)(

1 1 1

2

NdfXXSS

a

i

b

j

n

k

ijkT


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Tests of Hypotheses

1)()(1

2

1 1 1

2

adfXXbnXXSS

a

i

i

a

i

b

j

n

k

iA

a

i

b

j

n

k

b

j

jjB bdfXXanXXSS1 1 1 1

22 1)()(

Sum of Squares for factor A

Sum of Squares fo r factor B

Measures variation in the response due to the factthat different levels of factor A were used.

Measures variation in the response due to the fact

that different levels of factor B were used.


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Test of Hypotheses

Interact ion Sum of Squares

a

i

b

j

n

k

jiijAB badfXXXXSS1 1 1

2 )1)(1()(

Error or Residual Sum of Squares

a

i

b

j

n

k

ijijkE nabdfXXSS1 1 1

2 )1()(

Measures the variation in the response due to the

interaction between factors A and B. If the interactionplot is perfectly parallel this will be 0!

Measures the variation in the responsewithin the a x b

factor combinations.


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Tests of Hypotheses

So the Two-wayANOVA Identityis:

This partitions the Total Sum of Squares

into four pieces of interest for our

hypotheses to be tested.

EABBAT SSSSSSSSSS


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Tests of Hypotheses

As in One-wayANOVA, we obtain mean squares

for the different effects by dividing the sums ofsquares by their respective degrees of freedom

i.e.

These are our measures of variance for the analysis.

If an effect is not significant we expect

and if it is we expect

effect

effect

effect

df

SSMS

Eeffect MSMS

Eeffect MSMS


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Test of Hypotheses

F-Statistic for Testing an Effect

ondistributiFMS

MSF

E

effect

o ~Numerator df = dfeffect

Denominator df = dferror

If the F-statistic is large we reject that the effect is zero in

favor of the alternative that the effect of the factor is non-zero.


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Two-way ANOVA Table

Source of

Variation

Degrees of

Freedom

Sum of

Squares

Mean

Square F-ratio

P-value

FactorA a 1 SSA MSA FA =MSA /MSE Tail area

FactorB b 1 SSB

MSB

FB

=MSB

/MSE

Tail area

Interaction (a1)(b1) SSAB MSAB FAB =MSAB /MSE Tail area

Error ab(n1) SSE MSE

Total abn 1 SST

This is our initial focus

which is the p-value for

Question 1: Is there an

interaction effect?

T f H h


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Tests of Hypotheses

If the interaction is notstatistically significant

(i.e.p-value> 0.05) then we conclude the maineffects (if present) are independent of one another.

We can then test for significance of the main effectsseparately, again using an F-test.

If a main effect is significant we can then usemultiple comparison procedures as usual tocompare the mean response for different levels ofthe factor while holding the other factor fixed.

T f H h


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Tests of Hypotheses

If an interaction is significant (p-value < .05) we

conclude the main effects are not independent ofone another and that both effects are important!

In this case (i.e. the interaction is significant) thetests for main effects in the Two-way ANOVA

table are MEANINGLESS!

We must compare levels of factorA within

each levelof factor B (and vise versa).


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Example 1: Capsule Dissolve Time

Enter the n = 5

replicates for eachtreatment combination:Gastric, C

Gastric, V

Duodenal, C

Duodenal, V


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1st Highlight both

factors in this

list.

Next highlight

Full Factorial

from the Macros

pull-down menu.

Then click Run Modelleaving everything else

unchanged.


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Lots of extra CRAP we dont need. Turn

off the plots as they are unnecessary when

considering two-way ANOVA. Also we

really only need to consider the Effect

Tests portion of the numeric output

initially.


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Capsulex Fluid Type Interaction

Treatment combination

means.

Interaction Plot

Because the interaction is statistically significant

we are interested in comparing fluid types for a

given capsule type or comparing capsules for a

given fluid type.

The table on the right contains the results of all

pair-wise treatment mean comparisons,

however we are only interested in those as

described above.

Here we find that there is a significant

difference in the fluid types for the type C

capsules however there are no significant

differences between the capsules themselves for

given fluid type, nor is there a fluid effect whendissolving type V capsules.

We estimate that the mean time to dissolve type

C capsules in gastric fluid is between 3.57 and

23.43 minutes larger than the mean for

duodenal.


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Checking Assumptions

To check the assumptions of normality of the

response and equality of variance for the

difference treatment combinations we can

examine the residuals. For a two-wayANOVA the residuals are the deviations of the

observations from their respective treatment

combination sample means, i.e.

ijijkijk xxe


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To check the assumption of normality, assess the

normality of the residuals ijijkijk xxe

The residuals from thecapsule experiment

look approximately

normal with the

exception of twooutliers, but neither are

extreme enough to

warrant any concerns.


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To check the equality of variance for the

difference treatment combinations we can

examine the residuals plotted vs. the different

treatment combination means ijijkijk xxe

There appears to be more variation for the

dissolve times for type C capsules being

dissolved in gastric fluid. These

combination produced the two mildoutliers seen in the normal quantile plot.

Generally we worry when the variation

increases with the treatment combination

mean.


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Example 2: Comparing the Effectiveness of Three Forms ofPsychotherapy for Alleviating Depression

Suppose that a clinical psychologist is interested incomparing the relative effectiveness of three forms ofpsychotherapy for alleviating depression. Fifteenindividuals are randomly assigned to each of three

treatment groups: cognitive-behavioral, Rogerian, andassertiveness training. The Depression Scale of MMPIserves as the response. The psychologist also wished toincorporate information about the patients severity ofdepression, so all subjects in the study were classified ashaving mild, moderate, or severe depression. Thus wehave two factor of interest in this study: the treatmentthey received and the initial severity of their depression.


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Interaction Plot

Therapy Effect Plot

Degree of Severity Effect Plot


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Is there a significant interaction effect ? NO, p = .9717

Is there a significant therapy effect ?

Is there a significant degree of severity of effect ?

YES, p = .0356

YES, p < .0001

Now we can conduct multiple comparisons on each factor separately.


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We see that Rogerian

therapy differs significantly

from Cognitive-Behavioral

therapy, with Rogerian

having larger mean by

between .8 and 9.87 units.

We see that the initial severity of depression

levels significantly differ from each other in

terms of mean depression score. In particular

we see that those with a severe classificationhave a mean depression score exceeding that for

those with mild depression by between 8 and 18

points and those with moderate depression by

between 2 and 11.5 points.


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08.Two Way Anova