Question

Athletes and others suffering the same type of injury to the knee often require anterior and posterior ligament reconstruction. In order to determine the proper length of bone-patellar tendonbone grafts, experiments were done using three imaging techniques to determine the required length of the grafts, and these results were compared to the actual length required. A summary of the results of a simple linear regression analysis for each of these three methods is given in the following table. \({ }^{15}\) $$ \begin{array}{llrcc} \text { Imaging Technique } & \text {Coeffcient of Determination, } r^{2} & \text { Intercept } & \text { Slope } & p \text { -value } \\ \hline \text { Radiographs } & 0.80 & -3.75 & 1.031 & <0.0001 \\ \text { Standard MRI } & 0.43 & 20.29 & 0.497 & 0.011 \\ \text { 3-dimensional MRI } & 0.65 & 1.80 & 0.977 & <0.0001 \end{array} $$ a. What can you say about the significance of each of the three regression analyses? b. How would you rank the effectiveness of the three regression analyses? What is the basis of your decision? c. How do the values of \(r^{2}\) and the \(p\) -values compare in determining the best predictor of actual graft lengths of ligament required?

Short answer

Based on the given information from the three imaging techniques used to determine the proper length of bone-patellar tendon-bone grafts, rank their effectiveness and discuss the roles of \(r^2\) and p-values in determining the best predictor. The effectiveness ranking of the imaging techniques, based on their \(r^2\) values, is as follows: 1. Radiographs 2. 3-dimensional MRI 3. Standard MRI The coefficient of determination (\(r^2\)) measures how well the regression model fits the data. A higher \(r^2\) value indicates better predictions and is most appropriate for comparing the effectiveness of the imaging techniques in this problem. P-values measure the statistical significance of the regression. In this case, all p-values are less than 0.05, indicating that all three imaging techniques are statistically significant; however, p-values alone don't directly indicate their predictive power.

Step-by-step solution

Determine the Significance of the Regression Analyses

To determine the significance of each of the three regression analyses, we will look at their p-values. The smaller the p-value, the more statistically significant the analysis is. The p-values for the different methods are: - Radiographs: <0.0001 - Standard MRI: 0.011 - 3-dimensional MRI: <0.0001 Since all p-values are less than the common threshold of 0.05, we can say that each of the regression analyses is statistically significant.

Rank the Effectiveness of the Regression Analyses

To rank the effectiveness of the three regression analyses, we will take a look at the coefficients of determination (\(r^2\)). The higher the \(r^2\), the better the regression equation fits the data. The \(r^2\) values for the different methods are: - Radiographs: 0.80 - Standard MRI: 0.43 - 3-dimensional MRI: 0.65 Ranking the effectiveness based on the \(r^2\) values, we have: 1. Radiographs 2. 3-dimensional MRI 3. Standard MRI

Compare the Roles of \(r^2\) and P-Values

Coefficients of determination (\(r^2\)) and p-values have different roles in determining the best predictor of actual graft lengths. \(r^2\) is a measure of how well the model fits the data. A higher \(r^2\) implies a tighter fit of the regression line to the data points, and thus, better predictions. \(r^2\) values are best used to compare different regression models or methods, as we did in step 2. P-values, on the other hand, measure the statistical significance of the regression. A low p-value (typically below 0.05) indicates that the model is statistically significant, meaning that the observed results are very unlikely to occur by chance alone. P-values should be used to test the significance of a regression model or a certain predictor, but they are not useful for comparing models with different predictors. In this problem, relying on \(r^2\) values is more suitable for determining the best predictor of actual graft lengths since it is a measure of model fit. The p-values only serve to show that all the imaging techniques are statistically significant, but they do not directly indicate their predictive power.

Key concepts

Statistical Significance

In linear regression analysis, statistical significance reflects the likelihood that the relationship observed in the data set occurs by chance. When applying regression analysis to evaluate imaging techniques for graft measurements, the statistical significance is quantified using a p-value. A low p-value suggests that there's concrete evidence in the data set to infer that the relationship is real, not coincidental. For instance, both Radiographs and 3-dimensional MRI in the given table have p-values of less than 0.0001, indicating extremely strong evidence against the null hypothesis, which posits no relationship. Standard MRI has a p-value of 0.011, which is still below the commonly accepted significance level of 0.05, signifying it's statistically significant though less compellingly so than the others.

To enhance understanding, simplify the idea by equating statistical significance to a test's capability to determine if a particular finding is trustworthy. Think of it as evidence. If the p-value is under 0.05, it's as if a detective has enough evidence to conclude there's a definite link, rather than just a coincidental pattern.

Coefficient of Determination

Consider the coefficient of determination, or the \(r^2\) value, as a measure of how much of the variance in the dependent variable can be predicted from the independent variable. Imagine having a basket of fruit with many varieties; a higher \(r^2\) would imply that you can accurately predict the proportion of apples in the basket based on its weight, for instance.

In our exercise, the \(r^2\) values for the radiographs, standard MRI, and 3-dimensional MRI are 0.80, 0.43, and 0.65, respectively. These values indicate the proportion of variance in the actual graft lengths that is explained by each imaging technique. You can think of \(r^2\) as a score that tells us how well the data fits a line or curve. A higher \(r^2\) indicates that the imaging technique can more accurately predict the required length of bone-patellar tendonbone grafts based on the regression line. Thus, radiographs take the lead with the highest \(r^2\), followed by 3-dimensional MRI and standard MRI.

P-value

The p-value is a crucial part of any statistical hypothesis testing. It measures the evidence against a null hypothesis. The lower the p-value, the greater the statistical evidence that you should reject the null hypothesis. Essentially, it asks, 'If the null hypothesis were true, how likely is it to observe a result as extreme as the one in your study?'

In our case, all p-values are less than 0.05, meaning we have strong evidence against the null hypothesis, which would suggest there's no relation between the imaging techniques and the accurate determination of graft length. It's like finding a signed letter when you're trying to prove someone was at a certain place — a very low p-value is similar to having such hard evidence, thus strengthening your case. For the students, remember that while a p-value can tell you how likely your results could occur by chance, it doesn't say anything about the importance or the size of an effect — different concepts that should not be confused with the p-value itself.