ST2238 Introductory Biostatistics Lab 1 Lab 1: Introduction to Minitab, paired & two-sample t-tests Q1: Effect of hypnosis in reducing pain Hypnosis is sometimes used as a non-medicinal form of pain relief. Price and Barbour1 recruited students to a study in which they were exposed to painfully high temperatures before and after hypnosis in order to assess whether hypnosis has any effect on their ability to withstand pain. Participants indicated their (subjective) amount of pain on a 15cm strip. These pre- and post-hypnosis measurements for eight participants are reproduced below. Subject 1 2 3 4 5 6 7 8 Before 6.6 6.5 9.0 10.3 11.3 8.1 6.3 11.6 After 6.8 2.4 7.4 8.5 8.1 6.1 3.4 2.0 (a) Why is a paired t-test more appropriate than a two-sample t-test? (b) Do the data look normally distributed? (c) Perform a paired t-test to assess the hypothesis that hypnosis has no effect. State your conclusions carefully. (d) What is a 95% confidence interval for the true difference in mean subjective pain score before and after hypnosis?2 (e) What population does this sample represent? (f) Why are the conclusions of this experiment dubious? Q2: Ability of insects to detect floral iridescence Iridescence is the change in colour of a surface when viewed from different angles. Whitney et al.3 performed a laboratory experiment to assess whether floral iridescence is used to attract pollinating animals, by making artificial flowers out of CDs that were either iridescent or not, placing sweet substances on the iridescent tulip-castings and bitter on the non-iridescent ones, and recording how frequently laboratory bees identified the iridescent cast. Whitney et al. report that as the ten bees gained experience, they were more likely to visit the iridescent casting. Specifically, they state that of the first 10 visits per bee, the mean and standard deviation of the number of visits to the iridescent casting were 4.7 and 1.6, resp.; these were 8.1 and 1.3 on the last 10 (of 80) visits. Assume the number of visits in each of these groups of ten are approximately normally distributed. 1 Price and Barbour, 1987, J. Abnormal Psychol. 96:46–51. This is actually a stupid question included just for practice. 3 Whitney, Kolle, Andrew, Chittka, Steiner and Glover, 2009, Science 323:130–3. 2 Created by Alex Cook, National University of Singapore (a) Assess the hypothesis that the standard deviations are the same in the “population” of na¨ıve and experienced bees. (b) Perform a two-sample t-test of the hypothesis that bees do not learn with experience to identify iridescence. (Note, the authors took pains to ensure it was colour alone that was guiding the bees.) (c) Can you think of a better test for analysing the results of the experiment than a two-sample t-test? Q3: Do food resources decrease with distance from barrier reefs? Shima4 was interested in the habitat of the six bar wrasse (Thalassoma hardwicke Bennett 1830), a marine reef fish that inhabits parts of French Polynesia. He measured the settler density, defined as the number of juvenile fish per “unit” of settlement habitat, at a series of sites at two distances (250m, 800m) from the reef crests, in order to assess whether the resources upon which the wrasse depends decrease with distance from the reef crest. The data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ st2238/data/thardwicke.txt (a) Do the data look normal? (b) Are the variances sufficiently similar that a twosample t-test can be performed? (c) If so, perform a two-sample t-test of the hypothesis that the mean settler density is the same at the two distances. (d) Again, if the variances are similar enough, construct a confidence interval for the true difference between mean settler densities. Hints: Minitab is fairly intuitive, especially if you’ve ever used a spreadsheet in the past. Functions can be called by going to the menu along the top. It is also possible to use a command line to call functions. This is very useful when you are doing some repetitive analyses, as you can copy and paste commands in without having to go through each menu. To activate the command line, go to Editor → Enable commands. Note that this also allows you to see the commands equivalent to the functions you call using the menus. Minitab has a few sub-windows. One is the Worksheet. Your data go here, with one datum per cell. Ensure the first element of the data is in row 1 (unlike excel, Minitab is fussy about this). Columns can be referred to by their column number (e.g. C1) or by a name if you assign one (this goes between the “C1” row and row 1). Another sub-window is the Session window. This will contain the output of some functions you run. It will also be where the command prompt will appear if you use commands. There is also a Project Manager Window (which I don’t know how to use) and graphs will appear in their own windows. A picture of these windows can be seen overleaf. • To start Minitab, Windows icon → All programs → Minitab solutions → Minitab 4 Shima (2001), Ecology 82:2190–9. Page 1 ST2238 Introductory Biostatistics 15 Statistical Software English • To import data from a text file, File → Other files → Import Special Test. Choose the columns to store the data in (this requires knowing how many columns are in the data), click ok and then browse for the file. If the data have headings, Format allows lines to be skipped (i.e. you can skip the top line containing the heading). • To evaluate normality graphically, you can do a histogram or what Minitab calls a probability plot. • To do a histogram, Graph → Histogram, click simple, and in the next pop-up, enter the columns to plot histograms of in Graph variables. You might like to display multiple histograms on the same graph in different panels with the same scale (on the x-axis) to get a feel for how the distributions vary. This can be effected by playing with the options when you specify the columns for the plot. Lab 1 • You can create summary statistics of your data using Stat → Display Descriptive Statistics. Ask me if any of the meaning of any of these isn’t clear. Often students feel very attached to these and lovingly reproduce them in reports. I advise you are very selective in reporting summary statistics. • You can save either the whole project or just an individual worksheet. To save the project, File → Save Project. You might like to save each week’s work as a separate project, perhaps with a new worksheet per question. • A very useful tool in Minitab is its calculator. To access this Calc → calculator. This allows you to manipulate the entries of one or more columns and store them in another column or constant. [Constants are called “K1”, etc., and may be viewed using the command line via print k1.] This is a bit like excel’s functions, except you don’t have to mess about with dragging and clicking. • To do a probability plot, Graph → Probability Plots, then choose single. The plot it makes will have a straight line superimposed upon it. The points should lie close to the line if the data are normally distributed. Minitab has also superimposed some 95% confidence intervals (really, it’s a [pointwise] confidence region): 95% of the points should fall within this region. If too many fall outwith the region, this is suggestive of nonnormality. • Graphs can be saved to be imported into reports. To do this, File → Save graph as. There is a Mintab format, which no other program is likely to be able to open. Save it in this format if you wish to edit it in the future inside Minitab. Otherwise, save as a jpeg or png. Unfortunately, Minitab like much other windows software does not give the option of saving as lossless vector graphics, such as postscript or pdf. • Graphs can be edited by double clicking on various parts of them. Personally, I hate the cream background Minitab creates and the bold fonts for axes labels. (If you want really good graphs, I recommend mastering R—but this involves a considerable investment in time.) • A paired t-test can be done via Stat → Basic Statistics → Paired t. • A two-sample t-test can be done via Stat → Basic Statistics → 2-sample t. Note: if you have raw data in two columns, use the use samples in different columns option. There is also an option to perform the test based on summary statistics, i.e. the means, standard deviations and sample sizes. Note: I recommend ticking the assume equal variances box. If you do not, Minitab will do the version of the test that does not assume equal variances, and the interpretation will not be clear cut. • An f-test of the equality of variances can be done via Stat → Basic Statistics → 2 Variances Created by Alex Cook, National University of Singapore Page 2 ST2238 Introductory Biostatistics Lab 2: Non-parametrics, gression ANOVA, re- Lab 2 st2238/data/burnout.txt (a) Do the standard deviations look similar enough to do an ANOVA? (b) Perform an ANOVA. Is there any evidence of a difference in burnout levels among the four groups of physicians? Q4: Sign and signed-rank tests Q7: Two-way ANOVA Catnip is commonly believed to be a drug that influences the behaviour of cats. A volunteer5 at an animal shelter performed a study on the effect of the plant on 15 cats, by measuring the number of negative interactions each cat made during two 15 minute windows: before and after administering the drug. The data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ st2238/data/catnip.txt (a) Do the data look normal? (b) Use the sign test to test the hypothesis that catnip has no effect on the behaviour of the cats. (c) Try also the Wilcoxon signed-rank test to test this hypothesis. Are your results consistent? (d) You may also wish to perform a paired t-test. What does the pattern of p-values tell you about the tests? (A question for all the ex-NSmen!) Levels of testosterone decrease during times of stress. This is particularly the case for soldiers. Morgan and colleagues8 measured testosterone in 12 soldiers during a military exercise in which the (American) soldiers were “captured” and “interrogated”. The data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ st2238/data/soldiers.txt (a) Look at the data. Which appears most prominent: variability in adrenaline between soldiers or variability between times? (b) Fit a two-way ANOVA using soldiers as blocks. Does adrenaline change over the course of the exercise? Does adrenaline differ between soldiers? (c) Perform the following inappropriate analysis: a one-way ANOVA without blocking on soldiers. How do your conclusions change about the changing adrenaline levels over the course of the experiment? Q5: Rank-sum test A liana is a kind of woody vine that grows in the tropics. Ecologists6 measured abundance of liana (in stems/ha) at randomly chosen plots in the Amazon. Plots were randomised such that 34 were near the edge of the forest (within 100m) and 34 were far from the edge. The question that arises is: is the density of lianas the same near the edge of the forest? The data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ st2238/data/liana.txt (a) Do the data look normal? (b) Perform a Wilcoxon rank-sum test of the hypothesis that liana density near the forest edge is the same as the density far from the forest edge. (c) You may also wish to try a two-sample t-test of the same hypothesis. Given your answer to the first part of this question, which test would you prefer? Q6: One-way ANOVA L´ opez-Castillo and colleagues7 were interested in the effect of stressful types of medicine on the burnout levels of physicians. They selected 25 medics from infectious disease (working mostly with AIDS patients), hæmophilia (also working with many AIDS patients), oncology and internal medicine wards in Spain, and measured their “burnout” level using the Maslach Burnout Inventory questionnaire. A summary of the data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ 5 Jovan (2000). Stats 27:25–7 Laurance et al. (2001). Ecology 82:32–9 7 L´ opez-Castillo et al. (1999). Psychother. Pshcyosom. 68:348–56 6 Created by Alex Cook, National University of Singapore Q8: Linear regression Elephants are one of the most important mammals, in part because they are the mammals least related to man9 . They are therefore relatively well studied. Being able to predict the gestation period of pregnant elephant cows is important in preserving their numbers. A team of zoologists10 investigated the relationship between the cranial-rump length and gestational age. Although they fitted a more complex model, we shall assume a linear relationship is appropriate. The data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ st2238/data/elephant.txt (a) Plot the data. Does it appear there is a linear relationship between cranial-rump length and gestational age? (b) Perform a linear regression. What are your best estimates of the slope and intercept? (c) If an elephant cow presented to you with an embryo of cranial-rump length 140cm, what would be your best estimate of the age of her embryo? Hints: • The sign, Wilcoxon signed-rank and Wilcoxon rank-sum tests can be found under Stat → Nonparametrics. 8 Morgan et al. (2000). Biol Pschyiatry 47:1889–901 Dawkins (2004) The Ancestor’s Tale—I strongly recommend this book! 10 Hildebrandt et al. (2007). Proc R Soc Lond B 274:323–31 9 Page 1 ST2238 Introductory Biostatistics Lab 2 • Minitab calls the Wilcoxon signed-rank test the “1 sample Wilcoxon”. • Minitab calls the Wilcoxon rank-sum test the “Mann–Whitney”. • ANOVA can be found under Stat → ANOVA. • If you wish to do comparisons between pairs, there is a Comparisons option in the ANOVA window. I think “Fisher’s” corresponds most closely to what we spoke about during class. • Regression can be found under Stat → Regression → Regression. • You may be interested in exploring the power calculations Minitab allows. These allow you to determine the minimum sample size needed to achieve a specified power for particular parameter values. Check it out! Created by Alex Cook, National University of Singapore Page 2 ST2238 Introductory Biostatistics Lab 3: Multiple regression, binomial test, Fisher’s exact test, and contingency tables Q9: Multiple regression Public health decision makers are often interested in spatial heterogeneities in disease patterns, as these patterns may indicate ways to improve care. A groundbreaking study was carried out on factors influencing the propensity to lunacy in Massachusetts in the mid19th century, led by Edward Jarvis (who was not entirely co-incidently president of the American Statistical Association)11 . Recorded are multivariate data from 14 counties, each of which has: • Number of lunatics Lab 3 deviations above the mean of the controls. This happened for 2 controls and 20 cancer patients. Use Fisher’s exact test to assess whether pleiotrophin is higher in cancer patients. Q12: Contingency tables Phenotypic traits are often correlated in spatially distinct subpopulations. The Badeners are an ethnic group in south-west Germany, in a region that is close to the border of the northern and southern European areas. Ammon collected data on hair and eye colour among Badeners14 in the late 19th century, when the German population was less homogenised than now. These are presented below: Hair: Brown Black Fair Red Eye: Brown 438 288 115 16 Grey/Green 1387 746 946 53 Blue 807 189 1768 47 Is there an association between hair and eye colour in this ethnic group? Does this give any biological insight? • Distance to nearest mental health centre Hints: • County population (thousands, actually from 1950) • County population density per square mile • Percent of lunatics cared for at home The data can be downloaded from http://courses.nus.edu.sg/course/stacar/internet/ st2238/data/lunatics.txt Using a multivariate linear regression, determine what the factors are that influence lunacy occurence. Based on your conclusions, what would you advise to a policy maker? • For the lunatics question, you may wish to start by normalising the number of lunatics by dividing by the county size. You may also wish to consider transforming distance to the nearest centre, perhaps by taking the reciprocal. Q10: Binomial test Phillips and Smith12 wished to investigate whether it was possible for terminally ill patients to postpone death until after some significant date. They studies a group of ethnic Chinese women living in California who died within 1w of the harvest moon viewing festival. If they were unable to postpone their death until after the festival, we would expect the same number of deaths before and after the festival date. Of 103 women in the study, 70 died after and 33 died before the festival. Does this provide evidence that death can be postponed? Q11: Fisher’s exact test Souttou and colleagues13 were interested in whether the growth factor pleiotrophin was raised in individuals with pancreatic cancer. They measured this in 69 individuals—41 with the cancer and 28 controls—and noted whether the level was more than two standard 11 For a more recent discussion, see Hunter (1987), Geograph Rev 77:139–56 12 Phillips & Smith, 1990, J Am Med Assoc 263:1947–51 13 Souttou et al, 1998, J Natl Cancer Inst 90:1468–73 Created by Alex Cook, National University of Singapore 14 Ammon, 1899, Zur Anthropologie der Badener, Jena: Fis- cher Page 1

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