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The hypothetical data below are from a toxicity study designed to measure the effectiveness of different doses of a pesticide on mosquitoes. The table below summarizes the concentration of the pesticide, the sample sizes, and the number of critters dispatched. $$ \begin{aligned} &\begin{array}{l} \text { Concentra- } \\ \text { tion }(\mathrm{g} / \mathrm{cc}) \end{array} & 0.10 & 0.15 & 0.20 & 0.30 & 0.50 & 0.70 & 0.95 \\ &\hline \begin{array}{l} \text { Number of } \\ \text { mosquitoes } \end{array} & 48 & 52 & 56 & 51 & 47 & 53 & 51 \\ &\begin{array}{l} \text { Number } \\ \text { killed } \end{array} & 10 & 13 & 25 & 31 & 39 & 51 & 49 \\ &\hline \end{aligned} $$ a. Make a scatterplot of the proportions of mosquitoes killed versus the pesticide concentration. b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the concentrations and fit the line \(y^{\prime}=a+b\) (Concentration). What is the significance of a positive slope for this line? c. The point at which the dose kills \(50 \%\) of the pests is sometimes called LD50, for "Lethal dose \(50 \% . "\) What would you estimate to be LD50 for this pesticide and for mosquitoes?

Step by step solution

01

Creating a Scatterplot

First, we plot the proportions of mosquitoes killed versus the pesticide concentrations on a scatterplot graph. The x-axis represents the pesticide concentration and the y-axis represents the proportions of mosquitoes killed.

02

Calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\)

Next, we calculate the proportion \(p\) which is \((\text{Number Killed}/\text{Total Number of Mosquitoes})\) for each concentration. We then transform each of these proportions using the logistic transformation \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) to make it suitable for linear regression.

03

Fitting the Line

We then fit a regression line \(y^{\prime}=a+b\,\text{(Concentration)}\), where \(a\) is the y-intercept and \(b\) is the slope.

04

Interpret the Slope

A positive slope means there's a positive relationship between the pesticide concentration and the proportion of mosquitoes killed. The higher the concentration of the pesticide, the higher the proportion of mosquitoes killed.

05

Estimating LD50

LD50 estimates the pesticide concentration that would kill 50% of the mosquitoes. This can be calculated by rearranging the logistic regression equation to solve for concentration when \(y^{\prime} = \ln \left(\frac{0.5}{1-0.5}\right)= 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Scatterplots

When we talk about scatterplots, we're referring to a type of graph that's pivotal in statistics for displaying the relationship between two continuous variables. In the context of a toxicity study, a scatterplot can visualize how different concentrations of a substance, like the pesticide in our example, affect a particular outcome—in this case, the number of mosquitoes killed.

Each point on a scatterplot represents an observation from the dataset, with the x-coordinate corresponding to the pesticide concentration and the y-coordinate to the proportion of mosquitoes killed. For students, this is a practical way to see trends or patterns within their data. It could reveal, for instance, a trend that higher concentrations are associated with more frequent mosquito fatalities. Crafting a scatterplot is the first crucial step in data analysis as it can guide us on what type of statistical methods to use going forward.

Logistic Regression Explained

Now, why would we use logistic regression in a toxicity study? Logistic regression is used when the outcome variable (the proportion of mosquitoes killed, in this case) is binary or categorical. It calculates the probability of a certain event occurring, like how likely mosquitoes are to die at each pesticide concentration.

To employ logistic regression, we first need to transform our raw proportions using the logistic function. This step is crucial to ensure the response variable behaves in a linear fashion relative to the independent variable (pesticide concentration), hence making it appropriate for linear regression analysis.

In our example, the logistic transformation makes the relationship linear so that we can fit a regression line of the form y' = a + b(Concentration). Here, a stands for the y-intercept which gives the baseline log-odds of mosquito death when the pesticide concentration is zero, and b stands for the regression coefficient indicating how the log-odds of death increase with each unit increase in pesticide concentration. A positive slope b in this context indicates that as the concentration increases, the likelihood of mosquito death increases.

Estimating the LD50

Understanding the LD50, or 'Lethal Dose, 50%', is crucial for interpreting toxicity studies like the one we've been discussing. LD50 represents the dose at which a substance is lethal to half of the sample population. Estimating the LD50 helps to gauge the potency and potential risk associated with a substance.

In toxicity studies involving pesticides, researchers are particularly interested in this value as it informs them about the dosage required to achieve the desired pest control without posing excessive risks to other organisms.

To estimate LD50 from logistic regression, after performing the necessary data transformation and fitting the linear model, we set the logit (that is the logistic transformed proportion of the outcome) to zero. This corresponds to a 50% kill rate by definition, hence the term LD50. We then solve for the corresponding pesticide concentration. It’s an intricate dance between biology and statistics that yields insights into the impact and safety of substances used in environments shared by pests and humans alike.