A (second-order) multiple regression model with interaction terms

A example in which the predictors do not interact

Is baby’s birth weight related to smoking during pregnancy?

• Sample of n = 32 births

• Response (y): birth weight in grams of baby

• Potential predictor (x1): smoking status of mother (yes or no)

• Potential predictor (x2): length of gestation in weeks

A first order modelwith one binary predictor

iiii xxy 22110

where …

• yi is birth weight of baby i

• xi1 is length of gestation of baby i

• xi2 = 1, if mother smokes and xi2 = 0, if not

and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.

Estimated first order modelwith one binary predictor

0 1

424140393837363534

3700

3200

2700

2200

Gestation (weeks)

Wei

ght (

gram

s)

The regression equation isWeight = - 2390 + 143 Gest - 245 Smoking

In what way do the predictors have an “additive effect” on the response?

• The effect of smoking on the mean birth weight is the same for all gestation lengths. (Exhibited by parallel lines.)

• The effect of gestation length on the mean birth weight is the same for smokers and non-smokers. (Exhibited by parallel lines.)

What are “additive effects”?

A regression model contains additive effects if the response function can be written as a sum of functions of the predictor variables:

112211 pp xfxfxfyE

22110 xxyE

For example:

An example where including “interaction terms” is appropriate

Compare three treatments (A, B, C) for severe depression

• Random sample of n = 36 severely depressed individuals.

• y = measure of treatment effectiveness

• x1 = age (in years)

• x2 = 1 if patient received A and 0, if not

• x3 = 1 if patient received B and 0, if not

Compare three treatments (A, B, C) for severe depression

A B C

706050403020

75

65

55

45

35

25

age

y

A model with interaction terms

iiiii

iiii

xxxx

xxxy

31132112

3322110

where …

• yi is treatment effectiveness for patient i

• xi1 is age of patient i

• xi2 = 1, if treatment A and xi2 = 0, if not

• xi3 = 1, if treatment B and xi3 = 0, if not

If patient received B (xi2 = 0, xi3 = 1):

If patient received A (xi2 = 1, xi3 = 0): 112120 ii xyE

iiiiiiiii xxxxxxxy 311321123322110

113130 ii xyE

If patient received C (xi2 = 0, xi3 = 0): 110 ii xyE

In what way do the predictors have an “interaction effect” on the response?

In what way do the predictors have an “interaction effect” on the response?

• The effect of treatment on the treatment’s effectiveness depends on the individual’s age. (Exhibited by non-parallel lines.)

• The effect of the individual’s age on the treatment’s effectiveness depends on the treatment. (Exhibited by non-parallel lines.)

What does it mean for two predictors “to interact”?

• In general, two predictors interact if the effect on the response variable of one predictor depends on the value of the other.

• A slope parameter can no longer be interpreted as the change in the mean response for each unit increase in the predictor, while the other predictors are held constant.

What are “interaction effects”?

A regression model contains interaction effects if the response function cannot be written as a sum of functions of the predictor variables:

211222110 xxxxyE

For example:

112211 pp xfxfxfyE

The estimated regression function


If patient received A (xi2 = 1, xi3 = 0): 11 33.05.47703.003.13.4121.6ˆ ii xxy

If patient received C (xi2 = 0, xi3 = 0):

103.121.6ˆ ixy

The regression equation isy = 6.21 + 1.03age + 41.3x2 + 22.7x3 - 0.703agex2 - 0.510agex3

11 52.09.2851.003.17.2221.6ˆ ii xxy


A B

C

706050403020

80

70

60

50

40

30

20

age

y

y = 47.5 + 0.33x

y = 6.21 + 1.03x

y = 28.9 + 0.52x

Recall the appropriateregression analysis steps

• Model building– Model formulation– Model estimation– Model evaluation

• Model use

Residuals versus fits plot

756555453525

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Fitted Value

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Residuals Versus the Fitted Values(response is y)

Normal probability plot

P-Value (approx): > 0.1000R: 0.9866W-test for Normality

N: 36StDev: 3.63376Average: -0.0000000

50-5

.999

.99

.95

.80

.50

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.01

.001

Pro

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RESI1

Normal Probability Plot

Is there a difference in the mean effectiveness for the three treatments?


If patient received A (xi2 = 1, xi3 = 0): 112120 ii xYE

iiiiiiiii xxxxxxxY 311321123322110

113130 ii xYE

If patient received C (xi2 = 0, xi3 = 0): 110 ii xYE

Test for identical regression functionsAnalysis of VarianceSource DF SS MS F PRegression 5 4932.85 986.57 64.04 0.000Residual Error 30 462.15 15.40Total 35 5395.00

Source DF Seq SSage 1 3424.43x2 1 803.80x3 1 1.19agex2 1 375.00agex3 1 328.42

0: 1312320 H

49.24

4.15

4/42.3288.803

F

F distribution with 4 DF in numerator and 30 DF in denominator x P( X <= x ) 24.4900 1.0000

Does the effect of age on the treatment’s effectiveness depend on treatment?


If patient received A (xi2 = 1, xi3 = 0): 112120 ii xYE

iiiiiiiii xxxxxxxY 311321123322110

113130 ii xYE

If patient received C (xi2 = 0, xi3 = 0): 110 ii xYE

Test for significant interactionAnalysis of VarianceSource DF SS MS F PRegression 5 4932.85 986.57 64.04 0.000Residual Error 30 462.15 15.40Total 35 5395.00

Source DF Seq SSage 1 3424.43x2 1 803.80x3 1 1.19agex2 1 375.00agex3 1 328.42

0: 13120 H

84.22

4.15

2/42.328375

F

F distribution with 2 DF in numerator and 30 DF in denominator x P( X <= x ) 22.8400 1.0000

Another example

A model with one qualitative predictor and two quantitative

predictors

Bird breathing habits in burrows?

• Experiment with n = 120 nestling bank swallows and n = 120 adult bank swallows

• Response (y): % increase in “minute ventilation”, Vent, i.e., total volume of air breathed per minute

• Potential predictor (x1): percentage of oxygen, O2, in the air the baby birds breathe

• Potential predictor (x2): percentage of carbon dioxide, CO2, in the air the baby birds breathe

• Potential predictor (x3): 1 if adult, 0 if baby

Primary research question

• Is there any evidence that the adult birds differ from the baby birds in terms of their minute ventilation as a function of oxygen and carbon dioxide?

A formulated model

iiiii xxxy 3322110

where …

• yi is percentage of minute ventilation

• xi1 is percentage of oxygen

• xi2 is percentage of carbon dioxide

• xi3 is type of bird (0, if baby and 1, if adult)


An aside

An example that illustrates the impact of leaving a necessary interaction

term out of the model

Suggests x is related to y?Suggests there is a treatment effect?

0 1

109876543210

10

5

0

x

y

A formulated model

iiii xxy 22110

where …

• yi is the response

• xi1 is the variable you want to “adjust for”

• xi2 is treatment (0 or 1)


Is x related to y?Is there a treatment effect?

The regression equation is y = 4.55 - 0.028 x + 1.10 group

Predictor Coef SE Coef T PConstant 4.5492 0.8665 5.25 0.000x -0.0276 0.1288 -0.21 0.831group 1.0959 0.7056 1.55 0.125...Analysis of VarianceSource DF SS MS F PRegression 2 23.255 11.628 1.23 0.298Residual Error 73 690.453 9.458Total 75 713.709

Source DF Seq SSx* 1 0.435group 1 22.820

23.1

458.9

2/435.082.22*

F

298.023.173,2 FP

The estimated regression functions

0 1

109876543210

10

5

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x

y

The residuals versus fits plot

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A more appropriately formulated model

iiiiii xxxxy 211222110

where …

• yi is the response

• xi1 is the variable you want to “adjust for”

• xi2 is treatment (0 or 1)

• xi1 xi2 is the “missing” interaction term



0 1

109876543210

10

5

0

x

y

The regression equation isy = 10.1 - 1.04 x - 10.1 group + 2.03 groupx


0 5 10

-2

-1

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Fitted Value

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Is x related to y?Is there a treatment effect?

The regression equation isy = 10.1 - 1.04 x - 10.1 group + 2.03 groupx

Predictor Coef SE Coef T PConstant 10.1401 0.4320 23.47 0.000x -1.04416 0.07031 -14.85 0.000group -10.0859 0.6110 -16.51 0.000groupx 2.03307 0.09944 20.45 0.000

S = 1.187 R-Sq = 85.8% R-Sq(adj) = 85.2%

Analysis of VarianceSource DF SS MS F PRegression 3 612.26 204.09 144.84 0.000Residual Error 72 101.45 1.41Total 75 713.71

Back to the bird example

A more appropriately formulated model

iiiiiiiiiii xxxxxxxxxy 3223311321123322110

where …

• yi is percentage of minute ventilation

• xi1 is percentage of oxygen

• xi2 is percentage of carbon dioxide

• xi3 is type of bird (0, if baby and 1, if adult)


The model yields two response functions

For baby birds (xi3 = 0):

211222110 iiiii xxxxyE

For adult birds (xi3 = 1):

21122232113130 iiiii xxxxyE

Is there a significant interaction between type and O2? between type and CO2?

between O2 and CO2? The regression equation isVent = - 18 + 1.19 O2 + 54.3 CO2 + 112 Type - 7.01 TypeO2 + 2.31 TypeCO2 - 1.45 CO2O2

Predictor Coef SE Coef T PConstant -18.4 160.0 -0.11 0.909O2 1.189 9.854 0.12 0.904CO2 54.28 25.99 2.09 0.038Type 111.7 157.7 0.71 0.480TypeO2 -7.008 9.560 -0.73 0.464TypeCO2 2.311 7.126 0.32 0.746CO2O2 -1.449 1.593 -0.91 0.364

S = 165.6 R-Sq = 27.2% R-Sq(adj) = 25.3%

Is there a significant interaction between type and O2? between type and CO2?

between O2 and CO2?

Analysis of Variance

Source DF SS MS F PRegression 6 2387540 397923 14.51 0.000Residual Error 233 6388603 27419Total 239 8776143

Source DF Seq SSO2 1 93651CO2 1 2247696Type 1 5910TypeO2 1 14735TypeCO2 1 2884CO2O2 1 22664

49.0

27419

3/22664288414735*

F

69.049.0233,3 FP

The residual versus fits plot

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Residuals Versus the Fitted Values(response is Vent)

Plot for adult birds

13 14

0

400

15

800

16 17

Vent

18O2

86

4 CO22

18 019

Plot for baby birds

13-200

0

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200

400

15 16 17 18

Vent

O2

400

600

86

4 CO22

18 019

Is there any evidence that the adult birds differ from the baby birds?

The regression equation isVent = 137 - 8.83 O2 + 32.3 CO2 + 9.9 Type

Predictor Coef SE Coef T PConstant 136.77 79.33 1.72 0.086O2 -8.834 4.765 -1.85 0.065CO2 32.258 3.551 9.08 0.000Type 9.93 21.31 0.47 0.642


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Fitted Value

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Residuals Versus the Fitted Values(response is Vent)

The normal probability plot

P-Value (approx): > 0.1000R: 0.9978W-test for Normality

N: 240StDev: 164.009Average: 0.0000000

4003002001000-100-200-300-400

.999

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Normal Probability Plot

Cost of including unnecessary terms in the model

For model with interaction terms: Analysis of Variance


For model with no interaction terms: Analysis of Variance


Another comment about multiple testing

• Be aware of impact of multiple testing, but don’t be so extreme about it that your hands are tied.

• The serious danger is when you have many predictors -- not a small number associated with specific research questions.

• You can test multiple parameters simultaneously to reduce the number of tests you have to perform.

• You can reduce individual test’s α level.

A (second-order) multiple regression model with interaction terms

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