1. Analysis of Frequencies
So far --> response variable (Y) - continuous
Now look at response variable that is discrete/categorical
eg presence-absence or yes-no
--> ask questions about frequencies in each category

2. Chi -Square Test
- compare
OBSERVED frequency of a trait to
EXPECTED frequency of a trait
- by counting
- based on randomness or theory

3. Testing for randomness
Let’s say we’re interested in the ratio of females to males in this class. Is the composition a RANDOM sample of the UWO population?
Ho: the class is a random sample of UWO students
HA: the class is not a random sample of UWO students

4. I will have two observed frequencies, one for each of females and males,
two expected frequencies, one for each of females and males.

5. Where do the frequencies come from?
Observed:
Count all females and males
Observed frequency of females
Count
Females 132
Males 76
Total 208
Observed frequency of males

6. Where do the frequencies come from?
Expected:
Let’s say females:males at UWO is 50:50, half are females and half are males.
So, in a RANDOM sample, expect 50% female and 50% male.
Observed Expected
Females 132
Males 76
Total 208
208*0.5=104

7. Observed Expected
Females 132
Males 76
Total 208
104
208

8. Therefore reject Ho, not a random sample of UWO students

9. Expected:
Let’s say females:males at UWO is 60:40.
So, in a RANDOM sample, expect 60% female and 40% male.
Observed Expected
Females 132
Males 76
Total 208
208*0.6=124.8
208*0.4=83.2

10. Observed Expected
Females 132
Males 76
Total 208
124.8
83.2
208
Therefore, do not reject Ho, the class is a random sample of UWO students

11. Testing to see if your data fit a theoretical distribution
Hardy-Weinberg expectations for Mendel’s Peas
Dihybrid cross testing for independent assortment of traits
smooth-yellow 9
smooth-green 3
wrinkled-yellow 3
wrinkled-green 1
Ho: The peas occur in a ratio of 9:3:3:1
HA: The peas do not occur in a ratio of 9:3:3:1

12. Observed
Count
smooth-yellow 152
smooth-green 53
wrinkled-yellow 39
wrinkled-green 6
TOTAL 250

13. Expected
Count Expected
smooth-yellow *250=
smooth-green *250=46.875
wrinkled-yellow *250=46.875
wrinkled-green *250=15.625
TOTAL 250
9:3:3:1 ---> 9/16 : 3/16 : 3/16 : 1/16
: : :

14. Observed Expected O-E
smooth-yellow
smooth-green
wrinkled-yellow
wrinkled-green
TOTAL

15. Therefore, reject Ho, the sample is not in H-W Eq’m

16. Often, collect data on more than one variable simultaneously
--> contingency table analysis
--> test for independence among the variables
2 X 2 Contingency Tables
2 categorical variables, each with two levels

17. R1 = sum of observed in Row 1
C1 = sum of observed in Column 1
C2 = sum of observed in Column 2
C3 = sum of observed in Column 3
Total = sum of all observed

18. Expected calculation /

19. For example,
A rare tree species can
be rooted in serpentine or non-serpentine soil
have pubescent or smooth leaves

20. Is leaf morphology independent of soil type?
Ho: Leaf type is independent of soil type
HA: Leaf type is not independent of soil type

21. Again, want to compare OBSERVED to EXPECTED

22. Expected
= 34*28/100

23. Expected
= 34*72/100

24. Expected
= 66*28/100

25. Expected
= 66*72/100

27. Therefore, do not reject Ho: leaf type is independent of soil

28. For example, varieties of tiger beetles found during four times of the year

29. Ho: The occurrence of tiger beetle colour types is not dependent upon time of year
HA: The occurrence of tiger beetle colour types is dependent upon time of year

31. Therefore, reject Ho, occurrence of tiger beetle colour type is dependent upon time of year

32. Logistic Regression
--> used to describe the relationship between
Discrete Y
Continuous X
Y --> usually dichotomous or binary

33. Examples of dichotomous Y variable
Presence - Absence
Alive - Dead
Male - Female
Red - Green
Scored as 0’s and 1’s

38. Y = (X)

41. #Cigs