Indicator Variables

1
Indicator Variables

• sex (male or female)
• time (day, evening and night)
• season (spring, summer, autumn, winter)

2
Dummy variables
3
Note

• Number of indicator variables is needed is always
  1 less than the number of levels.
• There is usually more than one way to set up the
  variables.
• The values 0 and 1 are always used.

4
Greenfinch data

• Weight of the greenfinch
• Wingspan of the bird
• Age 0 for birds less than 1, 1 for birds over 1
• Sex 0 for males, 1 for females

5
Fitting the equations

• Once indicator variables are created they are
  entered into the equation in the same way as
  continuous variables.

6
Minitab output

• The regression equation is
• weight - 1.27 0.335 wing - 0.289 age 0.998
  sex
•  
• Predictor Coef SE Coef T
  P
• Constant -1.267 8.442 -0.15
  0.881
• wing 0.33452 0.09525 3.51
  0.001
• age -0.2891 0.3934 -0.73
  0.464
• sex 0.9981 0.4683 2.13
  0.035
•  
• S 1.875 R-Sq 10.5 R-Sq(adj) 8.0
•  

7
Note

• R-sq is very low.
• This means the equation may not be very good for
  making predictions.
• BUT the model still gives us information about
  the effect of each of the variables on weight.
• Weight increases with wing length
• Female birds are, on average, heavier
• Age has no effect

8
Omitting age

• The regression equation is
• weight - 0.54 0.325 wing 0.968 sex
•  
• Predictor Coef SE Coef T
  P
• Constant -0.545 8.366 -0.07
  0.948
• wing 0.32541 0.09424 3.45
  0.001
• sex 0.9682 0.4655 2.08
  0.040
•  
• S 1.871 R-Sq 10.0 R-Sq(adj) 8.4

9
Equations

• For males sex 0
• weight -0.54 0.325wing
• For females sex 1
• weight -0.52 0.325wing 0.968
• weight 0.448 0.325wing

10
Plot of data
Females are 0.968 grams heavier on average than
males
11
Interaction terms

• Weight b0 b1wingspan b2 sex b3
  wingspansex
• Previous equation fitted two parallel lines, one
  for each sex.
• Including an interaction term fits two completely
  separate lines, different intercepts and
  different gradients.
• The interaction term is not significant for this
  data set.

12
To include interaction term

• Use CalcgtCalculator the create a column wingspan
  multiplied by sex.
• Enter the new column into the regression model as
  before.

13
Power station example

• Response variable abundance of a species of
  worm, logged
• Predictor variables year and time of year
• Time fitted as 1-6
• Quarters of seasons fitted using 3 indicator
  variables.

14
Time series plot
15
CalcgtMake indicator variables

• Quarter is 1, 2, 3 or 4
• Indicator variables will be
• q1 1 for quarter 1, 0 otherwise
• q2 1 for quarter 2, 0 otherwise
• Etc.

16
Fitting regression equation

• Only 3 of the 4 indicator variables are needed.
• Any 4 can be used. The one omitted can be thought
  of as being used as the reference group.
• E.g. in the following output quarter 1 is the
  reference group.

17
Output

• The regression equation is
• a 2.13 0.00050 t - 0.123 q2 - 1.01 q3 0.443
  q4
•  
• Predictor Coef SE Coef T
  P
• Constant 2.12900 0.07261 29.32
  0.000
• t 0.000500 0.006393 0.08
  0.939
• q2 -0.12300 0.08112 -1.52
  0.158
• q3 -1.01350 0.08187 -12.38
  0.000
• q4 0.44350 0.08311 5.34
  0.000
•  
• S 0.1144 R-Sq 96.9 R-Sq(adj)
  95.8

18
Note

• t is not significant no evidence of a trend.
• t can be omitted from the equation.
• q2 is not significant but q3 and q4 are.
• Log abundance in quarter 3 is, on average, 1.01
  units lower than the log abundance for quarter 1.
• For quarter 4 it is 0.443 units higher than for
  quarter 1.
• It is advisable to keep q2 in the equation
  otherwise the interpretation of the coefficients
  changes.

19
Analysis of covariance

• An alternative approach to modelling data with a
  combination of qualitative and quantitative
  variables.
• Uses an analysis of variance model with
  covariates.
• Conclusions should be the same.

20
Summary

• All variables qualitative
• Analysis of variance
• All variables quantitative
• Regression analysis
• Some variables qualitative, some quantitative
• Regression analysis or analysis of covariance