Chapter 7 – Binary or Zero/one or Dummy Variables

Dummy Variables – Example

Example – WAGE1 Data Set

We want to fit the model:

The term female is a dummy variable and takes into account the effect of female vs. male.

Example – WAGE1 Data SetWe want to fit the model:

The regression equation iswage = 0.623 - 2.27 female + 0.506 educ

Predictor Coef SE Coef T PConstant 0.6228 0.6725 0.93 0.355female -2.2734 0.2790 -8.15 0.000educ 0.50645 0.05039 10.05 0.000

S = 3.18552 R-Sq = 25.9% R-Sq(adj) = 25.6%

Example – WAGE1 Data SetWe want to fit the model:

Interpretation of fitted model:wage = 0.623 - 2.27 female + 0.506 educThe coefficient on female measures the average difference in hourly wage between a woman and a man (for this model).

Graphical Interpretation of Sex Effect

Example – WAGE1 Data SetInterpretation of fitted model:wage = 0.623 - 2.27 female + 0.506 educ

The wage differential between females and males of -2.27 dollars per hour is due to sex of the individual and, potentially, factors we have not controlled for.

Example – WAGE1 Data Set

Now, fit the model:

This allows us to test the hypotheses:Ho: -> mean wage same for the sexes

H1:-> mean wage different for the sexes

Ho:

H1:

The regression equation iswage = 7.10 - 2.51 female

Predictor Coef SE Coef T PConstant 7.0995 0.2100 33.81 0.000female -2.5118 0.3034 -8.28 0.000

Ho:

H1:

The alternative hypothesis is supported – there is a statistically significant difference in mean wage between the sexes, p-value ≈ 0

wage = 7.10 - 2.51 female

Estimated mean wage of males is $7.10 per hour

Estimate mean wage of females is $7.10 – 2.51 = $4.59 per hour

Example – WAGE1 Data Set

What is the effect (if any) on wage of a person’s marital status?

(Did you check your regression assumptions?)

Example – WAGE1 Data Set

Examine the effect on wage of a few of the other dummy variables in this data set.

Example – WAGE1 Data Set

What is the interpretation of a dummy variable if the response is log(y)?

Now, fit the model:Log(

Example – WAGE1 Data Set

lwage = 1.81 - 0.397 female

Predictor Coef SE Coef T PConstant 1.81357 0.02981 60.83 0.000Female -0.39722 0.04307 -9.22 0.000 About 40% decrease in hourly wage if individual is female!

Example – WAGE1 Data Set

How do we handle multiple dummy variables at once?

Consider the two variables married and female.

We have four categories: single male, single female, married male, and married female.

We have four categories: single male, single female, married male, and married female.

Need to create three new dummy variables

Marriedmale Marriedfemale Singlefemale

Single Male 0 0 0

Married Male 1 0 0

Single Female 0 0 1

Married Female 0 1 0

We have four categories: single male, single female, married male, and married female.

Need to create three new dummy variables

In order to create the dummy variables marriedmale, marriedfemale, and singlefemale, use the calculator and a nested and statement within an if statement in Minitab.

We have four categories: single male, single female, married male, and married female.

wage = 5.17 + 2.82 marriedmale - 0.602 marriedfemale - 0.556 singlefemale

Predictor Coef SE Coef T PConstant 5.1680 0.3614 14.30 0.000marriedmale 2.8150 0.4363 6.45 0.000marriedfemale -0.6021 0.4645 -1.30 0.195singlefemale -0.5564 0.4736 -1.18 0.241

S = 3.35181 R-Sq = 18.1% R-Sq(adj) = 17.6%

Did you check the assumptions of homoskedasticity and normality?

We have four categories: single male, single female, married male, and married female.

lwage = 1.52 + 0.427 marriedmale - 0.0797 marriedfemale - 0.132 singlefemale

Predictor Coef SE Coef T PConstant 1.52081 0.05099 29.83 0.000marriedmale 0.42668 0.06155 6.93 0.000marriedfemale -0.07974 0.06552 -1.22 0.224singlefemale -0.13164 0.06680 -1.97 0.049

S = 0.472836 R-Sq = 21.3% R-Sq(adj) = 20.9%

Did you check the assumptions of homoskedasticity and normality?

lwage = 0.321 + 0.213 marriedmale - 0.198 marriedfemale - 0.110 singlefemale + 0.0789 educ + 0.0268 exper - 0.000535 expersq + 0.0291 tenure - 0.000533 tenursq

Predictor Coef SE Coef T PConstant 0.3214 0.1000 3.21 0.001marriedmale 0.21268 0.05536 3.84 0.000marriedfemale -0.19827 0.05784 -3.43 0.001singlefemale -0.11035 0.05574 1.98 0.048educ 0.078910 0.006694 11.79 0.000exper 0.026801 0.005243 5.11 0.000expersq -0.0005352 0.0001104 -4.85 0.000tenure 0.029088 0.006762 4.30 0.000tenursq -0.0005331 0.0002312 -2.31 0.022

S = 0.393290 R-Sq = 46.1% R-Sq(adj) = 45.3%

Did you check the assumptions of homoskedasticity and normality?

Example – BEAUTY Data SetDo looks affect hourly wage?

Variable: looksHas five levels: 1, 2, 3, 4, 5

Make dummy variables where: 1, 2 – below average3 – average4, 5 – above average

Example – BEAUTY Data SetDo looks affect hourly wage?

Make dummy variables where: 1, 2 – below average3 – average4, 5 – above average

You only need two dummy variables: belowaverage and aboveaverage

Example – BEAUTY Data SetDo looks affect hourly wage?

Your conclusions? Did you check assumptions?

Example – BEAUTY Data SetDo looks affect hourly wage?

Now, run a separate analysis for females and for males.

Your conclusions? Did you check assumptions?

Dummy Variables and the Interaction Term

Consider the Wage1 data set.

Response: log(wage)Predictor variables: female, married, female*married, educ, exper, exper^2, tenure, and tenure^2.

NOTE: need to run this model in general linear regression of Minitab

Dummy Variables and the Interaction Term

Term Coef SE Coef T PConstant 0.321378 0.100009 3.2135 0.001female -0.110350 0.055742 -1.9797 0.048married 0.212676 0.055357 3.8419 0.000female*married -0.300593 0.071767 -4.1885 0.000educ 0.078910 0.006694 11.7873 0.000exper 0.026801 0.005243 5.1118 0.000expersq -0.000535 0.000110 -4.8471 0.000tenure 0.029088 0.006762 4.3016 0.000tenursq -0.000533 0.000231 -2.3056 0.022

Summary of Model

R-Sq = 46.09% R-Sq(adj) = 45.25%

Dummy Variables and the Interaction Term

lwage = 0.321378 - 0.11035 female + 0.212676 married + 0.0789103 educ + 0.0268006 exper - 0.000535245 expersq + 0.0290875 tenure -0.000533142 tenursq - 0.300593 female*married

Interpretation of dummy variables coefficients.

Example – Problem C7.2

Complete C7.2 (i), (iii), and (iv) in class