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Problems of a significance test with a low power.

Reproducible research findings are a cornerstone of the scientific method. However, studies have shown that results of published research can be difficult to replicate when only statistically significant results are published, particularly when the statistical significance tests are conducted with small sample sizes (hence with low statistical power). Let’s consider a one sample t-test with Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-10Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-2 versus Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-1aProblems of a significance test with a low power. Reproducible research findings are a cornerstone...-4

(a) If the population standard deviation is 0.5, what is the power of the test when the effect size is δ = 0.1 and sample size is Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-51 = 10?

(b) In order to reject the null hypothesis, how large must the sample mean Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-6 be?

(c) If you rejected the null using data from one experiment, how likely can the statistically significant result be verified if you repeat the same experiment?

(d) Suppose that you realized that the test has a low power and you decided to repeat the experiment with a much larger sample size (e.g., Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-52 = 100). What is the power of the new test?

(e) Assuming δ = 0.1, how likely is it that you will obtain a sample mean as large as the statistically significant result from the experiment with Problems of a significance test with a low power. Reproducible research findings are a cornerstone...-51 = 10?

 
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Eutrophication due to increased input of nitrogen in the Neuse River Estuary in eastern North Carolina, USA, was considered the primary cause of large scale fishkills in the late 1990s. The North Carolina General Assembly established laws to protect the estuary, including a requirement of reducing nutrient (particularly, nitrogen) input to the estuary. Because eutrophication in North Carolina is measured by the concentration of chlorophyll a, assessing the success of the nutrient reduction program relies on the demonstration of a reduction in chlorophyll a concentration in the estuary. Three institutions have water quality monitoring programs including chlorophyll a in the variables they measure: NC Division of Water Quality (DWQ), University of North Carolina Institute of Marine Sciences (IMS), and Weyerhaeuser Corp. (WEY). Because the estuary is large and chlorophyll a concentrations vary spatially, methods used in sampling and measuring can affect the reported chlorophyll a concentrations. To demonstrate the success of the nitrogen reduction program, we need to compare the chlorophyll a concentrations measured before and after the implementation of the program to the concentrations before. But which series of data should we use? To answer this question, we need to compare the reported chlorophyll a concentrations from the three institutions and determine whether they are different. If they are the same, we may want to combine the three sources of data to increase the power of any statistical test we will use. If they are different, we need to describe the nature of the difference and decide how to best use them to describe the effect of the nitrogen reduction program.

For this problem, we are to compare the chlorophyll a concentrations from the three institutions and discuss the differences among them:

• Exploratory data analysis – a summary of data distributions and potential problems with the data.

• A decision on whether a transformation is necessary. In general, we use log-transformation for environmental concentration variables.

• ANOVA to test whether the mean (or median, if log-transformed) varies by institution.

• Present the estimated differences (and interpret the differences in the original concentration scale)

• A short discussion on other factors that may affect the result of this comparison.

 
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Harmel et al. [2006] compiled a cross-system data set to study the effects of agriculture activities on water quality. The data included in the study were mostly field scale experiments that measured nutrients (P, N) loading leaving a field. The data set (agWQdata.csv) includes the measured TP loading (TPLoad, in kg/ha), land use (LU), tillage method (Tillage), and fertilizer application methods (FAppMethd). You are to determine whether tillage methods affect TP loading.

(a) Estimate the mean TP loading for each tillage method (an easy way to do this in R is to use the function tapply):

Harmel et al. [2006] compiled a cross-system data set to study the effects of agriculture activities...

(b) Discuss briefly whether logarithm transformation is necessary.

(c) Use statistical test to study whether different tillage methods resulted in different TP loading (state the null and alternative hypothesis, conduct the test, report the result).

(d) Discuss briefly how useful is the test result.

 
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Huey et al. [2000] studied the development of a fly (Drosophila subobscura) that had accidentally been introduced from Europe (EU) into North America (N.A.) around 1980. In Europe, characteristics of the flies’ wings follow a “cline” – a steady change with latitude. One decade after introduction, the N.A. population had spread throughout the continent, but no such cline could be found. After two decades, Huey and his team collected flies from 11 locations in western N.A. and native flies from 10 locations in EU at latitudes ranging from 35 to 55 degrees N. They maintained all samples in uniform conditions through several generations to isolate genetic differences from environmental differences. Then they measured about 20 adults from each group. The data set flies.txt shows average wing size in millimeters on a logarithmic scale.

(a) In their paper, Huey et al. used four separate regression models to suggest that female flies from both EU and N.A. have the same wing length – latitude relationship (identical slopes), while the same relationships for male flies from the two continent are close but they were unable to say whether the slopes are the same.

We know that we can create a categorical variable to identify a fly’s origin and sex. This variable can be created by pasting the columns Continent and Sex:

Huey et al. [2000] studied the development of a fly (Drosophila subobscura) that had accidentally...-1

we obtain a model with four intercepts and four slopes, and the intercept and slope for the first level of FlyID (sorted alphabetically) is estimated and presented as the baseline.

Fit the linear model and interpret the results. Compare your results to the results presented in Huey et al. [2000]. Comment on any differences and why you feel you should use the approach we used here.

(b) The model we fitted here has its limitation. Only the slope and intercept of the first level are presented in the results explicitly. In this case, we will only see the intercept and slope for Female.EU, the baseline. Intercepts and slopes for the other three levels are presented in terms of their differences from the baseline. This is set up for hypothesis testing. That is, we can compare whether the slopes for Female.N.A., Male.EU, Male.N.A. are different from the slope for Female.EU. For this particular model, we can directly test whether the difference in slope between Female.EU and the slope of Female.N.A. is different from 0, but we cannot directly compare the slopes and intercepts for Male.EU and Male.N.A. To make this comparison, we must set Male.EU as the baseline first:

Huey et al. [2000] studied the development of a fly (Drosophila subobscura) that had accidentally...-2

which will change FlyID into a numeric variable with integers 1 to 4, and 1 is “Male.EU”, 2 is “Male.N.A.”, 3 is “Female.EU”, and 4 is “Female.N.A.”. Now refit the same model as in (a). Using results from both (a) and (b) to compare whether the slope for male flies from N.A. differs from the slope for male flies from EU, and whether the slope for female flies from N.A. differs from the slope for female flies from EU.

(c) In their paper, the linear regression models have very low R2 values, and the model we fit has a very high R2 value. Why? Is our model that much better?

 
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