Analysis of Environmental Data Problem Set Conceptual Foundations: Hy p o th e sis te stin g c o n c e p ts

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1 Analysis of Environmental Data Problem Set Conceptual Foundations: Hy p o th e sis te stin g c o n c e p ts Answer any one of questions 1-4, AND answer question 5: 1. Consider a hypothetical study of wood frog larval abundance in vernal pools. Let s say that you sample a vernal pool by taking 10 dip net sweeps of the water column in random locations throughout the pool and record the presence/absence of wood frog tadpoles in each sweep. Let s assume that the probability of tadpole capture is the same for every sweep. Given the probability distributions shown here, answer the following questions: a. Given the null hypothesis that larval abundance, expressed as the proportion of successful dip net sweeps, is distributed binomially with a trial size of 10 and a per trail probability of success of 0.3, what is the p-value associated with observing 5 or more successes out of 10 dips, and how did you derive your answer? b. Given (a) above, if you had set alpha=0.05 before the experiment, would you reject or fail to reject the null hypothesis, and why? c. Given (a) above and alpha=0.05, what is the Type II error rate; i.e., the probability wrongly failing to reject the null hypothesis? d. Given (a) above and alpha=0.05, what is the power of this test; i.e., the probability of correctly rejecting the null hypothesis? 2. Consider a hypothetical study on the affect of road crossing mortality on the age structure of spadefoot toad populations. Spadefoot toads typically undergo annual migrations to and from their breeding sites (seasonal ponds) and have extremely high fidelity to their breeding site (i.e., local populations are relatively independent). You hypothesize that road mortality is sufficient to affect population age structure, since at least a portion of the local population adjacent to a road would be subject to increased mortality rates due to road kill during migration to and from the breeding pond. Let s say you sample three local populations at their breeding ponds; one pond is adjacent to a busy highway, another one is next to a secondary road with moderate traffic rates, and another one is without any nearby roads. For each population, you randomly sample 100 individuals and determine how many years they each survive. Thus, the data represent for each individual (observation) the age

2 Hypothesis Testing Concepts: Problem Set 2 at death. The observed data are plotted here as a bar chart depicting the number of individuals surviving to each age for each of the populations. You are interested in knowing if the annual survival rate differs among the populations and estimating the probability of individuals in each population surviving for 10 years. Consequently, you specify the deterministic model to be an indexed vector of mean annual mortality rates (i.e., a vector consisting of three different mean mortality rates corresponding to the three different populations), and a geometric error for stochastic component. Based on this information, answer the following questions: b. Let s say that you want to test whether the observed differences in population age structure are significant. To do this you construct two versions of the model, one that assumes that there is a single age structure for all populations (null model) and one that allows the age structure to vary among populations (full model). You confront each of these models with the data and compute an objective measure of model fit for each. Lastly, you construct a test statistic called deviance to evaluate the difference in fit between models. Deviance is distributed chi-square with 2 degrees of freedom in this case. How you would go about calculating a p-value to determine if the test statistic (and hence the difference in fit between models) is significant? c. Let s say that you calculate the p-value to be Based on a conservative Type I error rate of 1% (i.e., not allowing more than a 1% chance of a Type I error), what would be your decision: reject or fail to reject the null hypothesis, and why? 3. Consider a hypothetical study on the affect of fire size on the severity of the fire. There is a general belief among land managers in the west that larger fires are more severe in terms of their ecological impacts. Fire severity is generally defined in terms of the proportion of the overstory vegetation killed by the fire and is often categorized into high severity, mixed severity and low severity. The belief is that as fires get larger the proportion of the fire (inside the fire perimeter) that is classified as high severity increases. Some claim this to be a myth. You decide to test this hypothesis. Specifically, you decide to test the hypothesis that the proportional extent of high severity burn increases linearly as fire size increases. To

3 Hypothesis Testing Concepts: Problem Set 3 do so, you specify the deterministic model to be a linear function of fire size (i.e., proportion high severity=a+bx) and assume normal errors for the stochastic component. To confront this model with data, you compile data on 100 randomly selected fires that occurred during the past 10 years in the Rocky Mountains region. For each fire you determine the size (ha) and proportion of high severity burn (via analysis of pre- and post-fire satellite images). The data are shown here as a scatter plot. Based on this data set, answer the following questions: b. Let s say that you fit the model to the data and estimate an intercept of 0.28 and a slope of 2e-06 (the solid line in the scatterplot). However, let s say that you don t trust the parametric p-value that your software spit out, so you decide to compute one yourself using Monte carlo randomization procedures. To do this, you simulate datasets that have a similar distribution of fire sizes as your original dataset, but without any relationship between size and proportion of high severity, which you specify as a linear model with an intercept equal to the mean proportion of high severity in the original dataset and a zero slope. You simulate 1,000 data sets and each time estimate the slope of the relationship. The figure shown here depicts the empirical cumulative probability distribution of the simulated slope values. The red dashed lines are drawn at 0.025, 0.05, 0.95, and Based on this distribution, what would be the approximate upper one-sided (or one-tailed) p-value for a slope of 2e- 06 (the original slope)? What would the approximate p-value be for a two-sided test? How did you derive your answers? c. Based on (b) above, would you reject or fail to reject the null hypothesis if the alternative hypothesis was that the slope differed from zero by at least 1.5e-06 given an a priori alpha=0.05? 4. Consider the following real study on the effect of three tree pruning methods on drag (a measure of stress on the tree stem caused by wind of a fixed speed). The data are shown here as a box-andwhisker plot. To test the null hypothesis of no effect of pruning method (type) on drag, you conduct a one-way analysis of variance, in which you derive a test statistic F that measures the ratio of among-pruning method to within-pruning method variance in drag and is distributed F with numerator degrees of freedom (df1) equal to 2 (number of methods minus 1) and denominator degrees of freedom (df2) equal to 232 (number of observations minus number of methods). Note, if

4 Hypothesis Testing Concepts: Problem Set 4 there is no variability in drag among methods (i.e., the means are equal), then all the variability in drag will be within method and the F statistic will equal zero. As the variance ratio increases, more of the variance in drag is due to difference among methods. At some point the ratio is large enough that one might conclude that drag is significantly different among methods and that the null hypothesis of no difference can be rejected. The quantile distribution for the F statistic with df1=2 and df2=232 is shown in the accompanying figure. Based on this information, answer the following questions: b. Let s say that you set alpha=0.1 and you observe an F=2.0, would you reject or fail to reject the null hypothesis, and why? c. If the observed value of F was 3.1, what is the approximate p-value for this test? 5. Consider a hypothetical study on the relationship between invasive earthworms and distance to agricultural land use. Let s say that you sample 100 plots in forest at varying distances to agriculture, ranging from 0-1,000 m, and at each plot you count the number of earthworms detected in a set of subplots using a liquid mustard solution that entices earthworms to the surface. The data are shown below as a scatterplot. Based on this information answer the following questions: a. What would a plausible statistical model be for this dataset? Note, be sure to consider both the deterministic and stochastic components, and justify your answer. b. Now, let s say you fit a simple linear model with normal errors (which may not be the best model specification) and estimate the intercept and slope of the linear model to be a=17.04 and

5 Hypothesis Testing Concepts: Problem Set 5 b= -0.01, respectively (depicted as the fitted line in the scatterplot). Let s say that you are interested in computing a p-value for the slope coefficient under the null hypothesis that the slope is zero. To do this, you convert your slope estimate into a t- statistic by dividing your estimate by its standard error (an estimate of sampling precision). In doing so, you can use the t distribution to derive a p-value. Recall that the t distribution is one of the distributions we use for sample statistics. The t probability distribution for 98 degrees of freedom (n-the number of model parameters) is shown in the figure below along with a shifted distribution representing the alternative hypothesis. In this case, the alternative is shifted by an amount equal to the observed t statistic for the dataset. Let s say that you are interested in a lower one-sided alternative hypothesis; i.e. that you want to test whether the slope is significantly less than zero (negative). An alpha=0.05 for a lower one-sided test is depicted by the shaded area under the null distribution. Based on this information, how would you graphically depict the power of this test? c. Let s say that you wanted to conduct a two-sided test instead. Would the power of the test increase or decrease, and why? d. How would the power of the test change if the slope estimate was even more negative, all other things equal, and why, and how would you graphically depict this change on the accompanying figure? e. How would the power of the test change if the sample size increased, and why, and how would you graphically depict this change on the accompanying figure?