AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

1. AAEC 4302ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH Chapter 7.3 Dummy Variables

2. Use of Dummy Variables (7.3) • In many models, one or more of the independent variables is qualitative or categorical in nature • This means that they can only take on few values which magnitude has little or no meaning

3. Use of Dummy Variables • This type of independent variables have to be modeled through dummy variables • A set of dummy variables is created for each categorical independent variable X in the model, where the number of dummy variables in the set equals the number of categories in which that independent variable is classified

4. Use of Dummy Variables • In our biological example is the skull length (mm) of the ith mouse: • X1i sex: male or female (two categories), • X2i specie (three categories), and • X3 age. • Two dummy variables will be created for X1 (D11 and D12) and three for X2(D21, D22, and D23)

5. Use of Dummy Variables • In the ith observation (mouse): • , if sex is male, 0 otherwise; • , if sex is female, 0 otherwise; • , if specie 1, 0 otherwise; • , if specie 2, 0 otherwise; • and ( , if specie 3, 0 otherwise. X1 X2

6. Use of Dummy Variables • The estimated model would be: • Notice that the dummy variables corresponding to the last categories of X1 and X2 (D12 and D23) have been excluded from the estimated model (any one dummy/category can be excluded, it makes no difference) • If you don’t exclude a dummy variable from a group, it will contain redundant information.

7. Use of Dummy Variables • Notice that this model actually estimates a different intercept for each observed sex/specie combination, while maintaining the same slope parameters for each of the other independent variables in the model ( ) (only one -age or - in our example)

8. Use of Dummy Variables Model to estimate: Estimated Model:

9. Use of Dummy Variables • For a male mouse of the first specie: 1 1 0 D11: 1 if sex = Male, 0 otherwise D21: 1 if species = 1, 0 otherwise D22: 1 if species = 2, 0 otherwise

10. Use of Dummy Variables • For a male mouse of the second specie: 1 0 1 D11: 1 if sex = Male, 0 otherwise D21: 1 if species = 1, 0 otherwise D22: 1 if species = 2, 0 otherwise

11. Use of Dummy Variables • For a male mouse of the third specie: 1 0 0 D11: 1 if sex = Male, 0 otherwise D21: 1 if species = 1, 0 otherwise D22: 1 if species = 2, 0 otherwise

12. Use of Dummy Variables • For a female mouse of the first specie: 0 1 0 D11: 1 if sex = Male, 0 otherwise D21: 1 if species = 1, 0 otherwise D22: 1 if species = 2, 0 otherwise

13. Use of Dummy Variables • For a female mouse of the second specie: 0 0 1 D11: 1 if sex = Male, 0 otherwise D21: 1 if species = 1, 0 otherwise D22: 1 if species = 2, 0 otherwise

14. Use of Dummy Variables • For a female mouse of the third specie: 0 0 0 D11: 1 if sex = Male, 0 otherwise D21: 1 if species = 1, 0 otherwise D22: 1 if species = 2, 0 otherwise

15. Use of Dummy Variables • Notice, therefore, that: • measures the difference in skull length (for any specie and age) between male and female mice • measures the difference in skull length (for male or female mice of any age) between species one and three

16. Use of Dummy Variables • measures the difference in skull length (male or female mice of any age) between species two and three • measures the difference in skull length (for male or female mice of any age) between species one and two

17. Use of Dummy Variables • measures the skull length for a female mouse of specie 3 at birth, i.e. age = 0 • : The skull length for a female mouse of specie 3 at birth is 33 mm.

18. Use of Dummy Variables • measures the difference in skull length (for any age) between male and female for any specie • : means that regardless of age, a male mouse will have a skull length 3.05 mm larger than a female mouse

19. Use of Dummy Variables • measures the difference in skull length (for male mouse of any age) between species one and three • : means that a mouse of species 1 will have a skull length 4.9 mm smaller than a mouse of species 3, regardless of sex and age.

20. Use of Dummy Variables • measures the difference in skull length (male or female mice of any age) between species two and three • : means that a mouse of species 2 will have a skull length 0.22 mm smaller than a mouse of species 3, regardless of sex and age.

21. Use of Dummy Variables • measures the difference in skull length (for male or female mice of any age) between species one and two • (-4.9) – (-0.22) = -4.68 means the skull length for species 1 is 4.68 mm shorter than for species 2, regardless of age and sex.

22. Use of Dummy Variables • A model like the former assumes that sex or specie shift the skull length regression function at the origin, in a parallel fashion, for example: Male of Specie 3 Y (mm) Female of Specie 3 3.05 (age)

23. Use of Dummy Variables • For that reason, the dummy variables in models like the former are often called “intercept shifters”

24. Use of Dummy Variables • In the previous example, six different (intercepts are actually being estimated for: • Male of specie 1 • Male of specie 2 • Male of specie 3 • Female of specie 1 • Female of specie 2 • Female of specie 3

25. Use of Dummy Variables • However, a single slope parameter ( ) that assumes and measures the same skull length-age relation for all six sex-specie combinations is estimated