Statistical Analysis in Environmental Science: Hypothesis Testing and Regression Analysis, Exams of Environmental Science

Download Statistical Analysis in Environmental Science: Hypothesis Testing and Regression Analysis and more Exams Environmental Science in PDF only on Docsity! ESCI 340 BIOSTATISTICAL ANALYSIS Course Review rev09_key.pdf McLaughlin Part Two: Practice Questions; Answer Key Answers in arial font. 1 After struggling for weeks to stay awake in your 1pm Biostatistics class, you and your classmates surmise that perhaps the subject is not so boring after all, but that people tend to fall asleep after lunch as their blood sugar decreases. You decide to test the null hypothesis that eating lunch does not affect a person’s ability to stay awake. ( H0 1 2: µ µ= , where µ1 is mean time awake without eating lunch, and µ2 is mean time awake after eating lunch) For six weeks, you skip lunch before every other class. On days that you do eat before class, you eat a lunch consisting of the same variety of apple and a peanut butter and jelly sandwich made of the same ingredients. During each class, you record the length of time between the beginning of class and when you begin to feel drowsy. After six weeks [n1 = 6 (lunch) and n2 = 6 (no lunch)], you obtain a t-value of 2.4, which exceeds the critical value t0.05(2),10 = 2.228. You show your analysis to your study partner, but she declares that your test is invalid. She is correct. Explain why. Your data are pseudoreplicated, because you measured responses of the same person (yourself) multiple times and treated those measurements as independent observations. In this case, n1 = n2 = 1, so ν = n1 + n2 -2 = 0. Not valid to compare with t0.05(2),10. 2 Human use of Federal public lands in the US exceeds 500 million visitor days annually. The US population currently numbers slightly more than 298 million. Ignoring foreign visitors, mean visitation of Federal public lands by US residents is about 1.67 days per capita. Using members of this class as a sample, determine the probability that the mean visitation rate of WWU/Huxley environmental science students is the same as the national mean visitation rate. Answer using one-sample t-test, with 67.1:0 =µH . With n = 20, critical value: tα(2),n-1 3 As discussed on the first day of class, many questions in environmental science can be posed as "Is there a difference between sampled populations?", "Is there an effect of one variable on another?", or "Is there an association between two variables?" Using statistical tests in this course, answers to these questions depend on (1) the magnitude of the difference, effect, or association, and (2) sample variability. For each of the following tests, identify the variable representing each of these quantities. Example: two- sample t-test, magnitude of difference = 21 XX − ; sample variability = 21 XXs − Magnitude Variability a) Paired-sample t-test d ds b) Single factor ANOVA groups MS error MS c) Tukey multiple comparison test BA XX − nerrorMS /SE = d) Simple linear regression reg.MS or b – β0 residual MS or sb ESCI 340 BIOSTATISTICAL ANALYSIS Course Review: Answer Key rev09_key.pdf 2 McLaughlin 4 In 1999 the American Society for the Testing of Materials (ASTM) announced a change in standards for evaluating chemical toxicity. Previously, toxicity was determined using analysis of variance and associated multiple comparisons tests, in which test subjects were exposed to one of several concentrations of the compound of interest, including zero concentration. Results were used to determine concentrations that caused (statistically) significant effects and concentrations that had no detectable effect. The new standard uses regression analysis to determine dose-response relationships and to determine the threshold concentration below which no effects are observed. Which approach provides more statistical power? Why? Regression provides greater statistical power because it can be used to determine the threshold concentration precisely. Conclusions of analysis of variance are restricted to the exposure levels studied. If the sampling regime does not include the actual threshold exposure, analysis of variance (and associated multiple comparisons test) will identify the nearest (greater) exposure as the threshold. Hence, analysis of variance would miss effects at exposures between the actual threshold and identified (sampled) value. Overlooking these effects is a type II error, and reduces statistical power. For the following two questions (5 and 6), describe the following. (a) The appropriate statistical analysis to perform, including number of tails, fixed or random effects, and parametric or nonparametric tests. (b) State any assumptions necessary in using the appropriate statistical analysis. (c) State the null hypothesis or hypotheses to be tested. (d) State the criterion for rejection of the null hypothesis or hypotheses. 5 Research question: Is there an association between the depth of tidepools and number of microhabitats within them? Data on these two characteristics were recorded for 100 randomly selected tidepools. a) Simple linear correlation, two tailed, parametric b) Samples drawn from bivariate normal distribution. X & Y sampled at random from normally distributed populations. c) ρ = 0 d) t-test: t t≥ α ν( ),2 where ν = n − 2 = 98 F-test: F F≥ α ν ν( ), ,2 , i.e., 98,98),2(αFF ≥ 6 Research question: Do textbooks required for natural science courses cost more than texts required for social science courses? Prices charged for required texts were recorded from samples of ten courses each selected from upper and lower division courses in the natural sciences and social sciences (20 courses total). a) Two-sample t-test, one-tailed, parametric b) Normally distributed residuals c) ... socialsciscinat µµ ≤ d) If α = 0.05, 734.118),1(05.0 ),1( =≥ ≥ tt tt calc calc να ESCI 340 BIOSTATISTICAL ANALYSIS Course Review: Answer Key rev09_key.pdf 5 McLaughlin Transform predictor (independent) variable (ppt), using square root transformation ′ = +X X ½ , or logarithmic transformation )1log(' += XX . [sediment] (g/L) Log(ppt) 10 Research question: how is elk migration down to private land in autumn affected by date, start of hunting season, and number of hunters? The Colorado Division of Wildlife conducted an experiment in which hunting season dates and numbers of hunters allowed in experimental areas were manipulated. Elk migration in those areas was measured, and a set of models were fit to data on elk movement, with the following results. Model No. Model variables K AICc ∆AICc w 1 date, area, # hunters, huntseason, 2-way interactions 11 5835.37 0 0.441 2 date, area, # hunters, huntseason, 2- and 3-way interactions 15 5835.52 0.15 0.409 3 date, area, # hunters, huntseason, 2-, 3-, and 4-way interactions 16 5837.53 2.16 0.150 4 date, year, area, huntseason, 2-way interactions 19 5847.17 11.80 0.001 5 date, year, area, huntseason, 2- and 3-way interactions 29 5863.74 28.37 0.000 6 date, year, area, huntseason, 2-, 3, and 4-way interactions 32 5869.81 34.44 0.000 a) Complete the table above, by filling in values for ∆AICc and w. b) Which model performs best according to Information criteria? Model 1, because it has lowest AICc score. c) What is the probability that the model identified in (b) really is the best among the models considered? W1 = 0.441 or 44.1 % probability d) What is the confidence set for the best model, among the models considered? Method 1, 95%: Confidence set: [1, 2, 3] Method 2, for ∆≤2: Confidence set: [1, 2] Method 3, with C = 20: Confidence set: [1, 2, 3]. ESCI 340 BIOSTATISTICAL ANALYSIS Course Review: Answer Key rev09_key.pdf 6 McLaughlin 11 Is the allocation of Greenways money among geographic regions of Bellingham unfair relative to the number of people living in those regions? Treat the following information (Greenways expenditures from 1990 and 1997 levies; population data from 2000 census) as sample data to address the research question. Use Chi-squared goodness of fit test. Region % population % Greenways $$ (fi – f^)2/f^ Lower Wh. Cr. – Boulevard Pk. Corridor 9 3 4.000 NW Bellingham & Urban Fringe 26 27 0.036 SW Bellingham 30 41 4.033 SE Bellingham & Urban Fringe 10 7 0.900 NE Urban Fringe 2 3 0.500 East & NE Bellingham 23 19 0.696 17.102 =calcχ 07.11 2 5,05.0 =χ 0.10 > P > 0.05 In what form would you need this information to address the question more appropriately? Greenways expenditures as frequencies (actual $$ spent) ESCI 340 BIOSTATISTICAL ANALYSIS Course Review: Answer Key rev09_key.pdf 7 McLaughlin 12 Do benzene and dioxin impact survival of fathead minnows to the same degree? Survival rates of 50 randomly selected fathead minnows were recorded after exposure to a range of benzene concentrations, and survival rates of 50 other randomly selected fathead minnows were recorded after exposure to a range of dioxin concentrations. Simple linear regression analysis produced the following results (some values omitted intentionally). Use these results to answer the research question. Benzene Exposure (fraction alive / ppm) a ( intercept) = 0.98 sa = 0.02 b (slope) = − 0.12 sb = 0.033 Source of variation DF SS MS Total 3.7 Regression 2.5 Residual 1.2 Benzene )( XXi − 22.6 Dioxin Exposure (fraction alive / ppm) a ( intercept) = 0.98 sa = 0.02 b (slope) = − 0.22 sb = 0.036 Source of variation DF SS MS Total 4.1 Regression 2.7 Residual 1.4 Dioxin )( XXi − 22.6 Null hypothesis: H0: β1 = β2 21 21 bbs bbt − − = ( ) ( ) 2 2 2 1 2 2 )()( 21 ∑∑ ⋅⋅ − += x s x s s pXYpXYbb 21 212 )DF residual()DF residual( )SS residual()SS residual()( + + =⋅ pXYs 04.2 049.0 )22.0(12.0 = −−− =t t0.05(2),96 = 1.985 0.05 > P > 0.02 If α = 0.05, reject H0 Conclude that dioxin decreases survival more than does benzene.