Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture #3 Review: Analysis of Count Data

Noxious Weeds The following table reports the number of noxious weed seeds found in 98 samples of meadow grass. Determine a probability distribution that reasonably represents these data.

Skin Damage In a study of leprosy patients, each patient was classified at the beginning of the study according to whether they had a little or much infiltration (skin damage). At the end of the study, they were classified according to the change in their health. Is there a relationship between a patient’s skin infiltration and his/her improvement?

Effectiveness of Penicillin In an early experiment involving penicillin, 130 mice were injected with bacteria and either treated with penicillin or left untreated (controls). Do the following data indicate that treatment with penicillin is effective?

Alternatives to Penicillin In a study of possible alternatives to penicillin, mice were exposed to the same bacteria as in the previous study and then treated with vitamins Niacinamide (NA), folic acid (FA), p-amino-benzoic acid (Paba), or B 6 as pyridoxin. Can it be concluded that any or all of these treatments have a success rate different from 25% ?

Effectiveness of Penicillin The table below contains the combined results of the previous two experiments. Do these data indicate that there is any difference in the effectiveness of the various treatments?

Example #1 An experiment was conducted to determine whether reading to preschool children could influence their behavior. To conduct the experiment, 80 groups of 5 children were tested. Each group of children was read a story in which a young child was praised and rewarded for being polite and letting others be the first to go outside to play. Then, separately, each of the five children in the group was asked to go outside with other children to play. The number (out of five) in each of the groups who let the other children go outside first was then recorded and is shown in the following table, in the row labeled “Number Polite.” For example, 18 of the 80 groups had 3 children who were labeled “polite” because, after being read the story, they let the other children go out to play first. 1. Suppose these children can be considered a representative sample from a well-defined population. Estimate the overall proportion of children who are “Polite.” Obtain point and interval estimates. Write a short interpretation of the results. 2. Suppose that it is hypothesized that 50% of the children from this population, after being read the story, would be “polite.” Select a probability distribution that you feel is appropriate for the above grouped data and determine whether the data are in agreement with your assumed probability distribution. Make a complete analysis, including a written formal statistical conclusion.

Example #2 Geneticists have long been interested in the relationships, if any, in eye color of parents and siblings. The data below were collected to determine whether any relationships exist. Eye colors from grandfathers, fathers, and sons from a total of 3,245 families were obtained and are shown below. 1. Determine whether an association exists between a father’s eye color and a son’s eye color for those families whose grandfathers had light eyes. Make a complete analysis, including a formal statistical conclusion. If an association exists, explain what it is. 2. Is the same conclusion that you drew for the light-eyed grandfathers valid for the dark-eyed grandfathers? Be complete in your analysis. 3. What assumptions did you make in order to perform the analyses? 4. For those families whose grandfathers had light eyes, calculate the odds for light-eyed vs. dark-eyed sons for (a) the light-eyed fathers and (b) the dark-eyed fathers. Carefully interpret each of the odds and the odds ratio. Test whether the odds ratio is 1.

Example #2 Geneticists have long been interested in the relationships, if any, in eye color of parents and siblings. The data below were collected to determine whether any relationships exist. Eye colors from grandfathers, fathers, and sons from a total of 3,245 families were obtained and are shown below. 5. Assuming that the odds ratios are the same for the light-eyed and the dark-eyed grandfathers, determine whether the odds ratio across grandfather eye colors is different from Consider the combinations of grandfathers’ and fathers’ eye colors to be 4 distinct “ancestral” combinations of eye color. Is there an association between the son’s eye color and the combinations of grandparent/parent eye colors? If one exists, what is it?