**You must type this and all other homework assignments. Do not e-mail the assignment to me; turn it in early (at 322 Wolf) for a foreseeable absence, or turn it in late after an unexpected absence from class. **

**1. **Calculate the residuals for the data you collected for one-way anova. Remember that the residual in this case is the difference between an observation and the mean of its group. For example, if you measure the lengths of leaves on trees A, B, C, D, E, and F, the residual for leaf 1 on tree A would be the length of leaf 1 minus the mean length for tree A.

Next, put all the residuals in one column and draw a histogram of them, using the spreadsheet linked from the href="http://www.biostathandbook.com/normality.html">normality page. Remember that to copy and paste the numerical results of a formula, you should copy, then choose "Paste Special..." from the Edit menu. Then select the "Values" checkbox, so only the numerical values will be pasted.

Turn in the histogram of your untransformed residuals (just the graph, not the whole spreadsheet).

**2. **Log-transform your data set (the original data, not the residuals). Then calculate the mean of the log-transformed data for each group, and the residuals of the log-transformed data. Draw a histogram of the residuals of the log-transformed data.

**3. **Which looks more normal--your untransformed data, or your log-transformed data? Do you think it looks normal enough for an anova?

**4. **Using either my spreadsheet or SAS (you don't have to do both), do a one-way anova on your log-transformed data. Report the F-statistic and the P-value. Also report the F-statistic and P-value for your untransformed data (that you analyzed in the previous assignment). How do the results compare?

**5. **Download the spreadsheet for Bartlett's test linked from the homoscedasticity page. We didn't talk about it in class; all you need to know is that Bartlett's test has a smaller P-value when there's greater heteroscedasticity. Enter your one-way anova data (the raw data, not the residuals). Turn in the standard deviations for your groups and the P-value for the Bartlett's test. The spreadsheet gives you the option of log-transforming your data; do each, and turn in the standard deviations and P-value for the Bartlett's test for your log transformed data.

**6. **Did it look like your untranformed data had a lot of heteroscedasticity? Did the log transformation make it better?

**7. **Two technicians, "Brad" and "Janet" (these are real data, but I've changed their names), have measured the uptake of a fluorescently labelled protein in rat kidneys. They want to know whether their techniques for anesthetizing the rats, cutting them open, and injecting the fluorescently labelled protein are the same. They also want to know whether it's worthwhile to use multiple rats, and whether it's worthwhile to make multiple measurements from each rat. Brad operates on three rats, then after 60 minutes, measures the protein in 10 samples from each rat. Janet does the same on three rats of her own. The data are at the bottom of this page. Analyze the data using a nested anova. Report the F-statistics and P-values for each null hypothesis. You must use my spreadsheet, and you must also analyze the data using SAS. If you get the same P-values in SAS as you did with the spreadsheet (note that SAS may round to a smaller number of decimals), you only need to print the SAS output (from the .lst file). If you get a different result, also print the .log file and your SAS program (your .sas file).

**7. **interpret the results of question 6--what would you recommend to Brad and Janet?

Tech Rat Protein Janet 1 1.119 Janet 1 1.2996 Janet 1 1.5407 Janet 1 1.5084 Janet 1 1.6181 Janet 1 1.5962 Janet 1 1.2617 Janet 1 1.2288 Janet 1 1.3471 Janet 1 1.0206 Janet 2 1.045 Janet 2 1.1418 Janet 2 1.2569 Janet 2 0.6191 Janet 2 1.4823 Janet 2 0.8991 Janet 2 0.8365 Janet 2 1.2898 Janet 2 1.1821 Janet 2 0.9177 Janet 3 0.9873 Janet 3 0.9873 Janet 3 0.8714 Janet 3 0.9452 Janet 3 1.1186 Janet 3 1.2909 Janet 3 1.1502 Janet 3 1.1635 Janet 3 1.151 Janet 3 0.9367 Brad 5 1.3883 Brad 5 1.104 Brad 5 1.1581 Brad 5 1.319 Brad 5 1.1803 Brad 5 0.8738 Brad 5 1.387 Brad 5 1.301 Brad 5 1.3925 Brad 5 1.0832 Brad 6 1.3952 Brad 6 0.9714 Brad 6 1.3972 Brad 6 1.5369 Brad 6 1.3727 Brad 6 1.2909 Brad 6 1.1874 Brad 6 1.1374 Brad 6 1.0647 Brad 6 0.9486 Brad 7 1.2574 Brad 7 1.0295 Brad 7 1.1941 Brad 7 1.0759 Brad 7 1.3249 Brad 7 0.9494 Brad 7 1.1041 Brad 7 1.1575 Brad 7 1.294 Brad 7 1.4543

Return to the Biological Data Analysis syllabus

Return to John McDonald's home page

This page was last revised October 18, 2013. Its URL is http://udel.edu/~mcdonald/stathw7.html