Question

An experiment was conducted to study the effect of tamoxifen on patients with cervical cancer. One of the measurements made, both before and again after tamoxifen was given, was microvessel density (MVD). MVD, which is measured as number of vessels per \(\mathrm{mm}^{2}\), is a measurement that relates to the formation of blood vessels that feed a tumor and allow it to grow and spread. Thus, small values of MVD are better than are large values. Data for 18 patients are shown. \({ }^{23}\) How might these data be analyzed? $$ \begin{array}{|crr|ccc|} \hline & \text { MVD } & \text { MVD } & & \text { MVD } & \text { MVD } \\ \text { Patient } & \text { before } & \text { after } & \text { Patient } & \text { before } & \text { after } \\ \hline 1 & 98 & 75 & 10 & 70 & 60 \\ 2 & 100 & 60 & 11 & 60 & 65 \\ 3 & 82 & 25 & 12 & 88 & 45 \\ 4 & 100 & 55 & 13 & 45 & 36 \\ 5 & 93 & 78 & 14 & 159 & 144 \\ 6 & 119 & 102 & 15 & 65 & 27 \\ 7 & 70 & 58 & 16 & 98 & 90 \\ 8 & 78 & 70 & 17 & 66 & 16 \\ 9 & 104 & 90 & 18 & 67 & 53 \\ \hline \end{array} $$

Short answer

To analyze these data, calculate the difference in MVD before and after treatment for each patient, then use a paired t-test or a Wilcoxon signed-rank test based on the distribution of differences, and interpret whether the change is statistically significant.

Step-by-step solution

Understanding the problem

Recognize that to analyze the effect of tamoxifen on patients with cervical cancer using the given data, statistical methods must be employed to determine if there is a significant difference in MVD values before and after treatment.

Choosing an appropriate statistical test

Since we have paired data (MVD before and after treatment for the same patients), a paired t-test or a non-parametric test such as the Wilcoxon signed-rank test could be used to analyze the differences in MVD values depending on the distribution of the differences.

Calculating the differences

Calculate the difference in MVD for each patient by subtracting the 'after' value from the 'before' value. This will yield a new set of data points representing the change in MVD due to treatment.

Analyzing the distribution of differences

Plot the differences or calculate descriptive statistics to determine if the differences are normally distributed. If the differences are normally distributed, a paired t-test is appropriate; otherwise, a non-parametric test is preferred.

Applying the statistical test

Conduct the chosen statistical test. If using a paired t-test, calculate the t-statistic and compare it to the critical value for the chosen significance level. For a non-parametric test, follow the specific steps for that procedure.

Interpreting the results

Determine whether there is a statistically significant difference in MVD before and after treatment based on the p-value obtained from the test. A low p-value (typically less than 0.05) would indicate a significant effect of tamoxifen on reducing MVD.

Key concepts

Paired t-test

When scientists want to determine the effectiveness of a treatment, like tamoxifen on patients with cervical cancer, they often rely on what is called a paired t-test. This test is particularly useful for comparing two measurements taken from the same group under different conditions—in this case, microvessel density (MVD) values of patients before and after the treatment.

The key concept behind the paired t-test is that it is designed to account for the natural variability between subjects by pairing the data points. This means that each patient serves as their own 'control', reducing the impact of external variables that may differ from one patient to another.

To perform a paired t-test, one first calculates the differences between each patient's before and after measurements. Then, these differences are analyzed to check if they follow a normal distribution (which we'll discuss in a later section). If the differences do indeed follow a normal distribution, the paired t-test can be applied to determine if the average difference is significantly different from zero, which would suggest a treatment effect. It does this by generating a t-statistic—a ratio of the mean difference to the variability in the data—allowing the researcher to understand if the observed effect is larger than what might be expected by chance.

Wilcoxon signed-rank test

Sometimes, the differences we calculate in paired data do not follow a normal distribution. In those cases, we turn to a non-parametric alternative to the paired t-test—the Wilcoxon signed-rank test. This test is less sensitive to extreme values and does not assume that the data come from a specific distribution, making it more appropriate for skewed data or data with outliers.

The Wilcoxon signed-rank test involves ranking the absolute values of the differences between paired observations, without considering their signs. After ranking, each difference is given a sign corresponding to whether the post-treatment value was higher or lower than the pre-treatment value. The ranks are then used to calculate a test statistic that can determine whether there is a significant difference in the central tendency of the two related groups. This approach essentially measures whether the positive and negative differences are symmetrically distributed around the median, if not symmetric, it indicates a significant effect of the treatment.

Normal Distribution

A normal distribution, also known as a Gaussian distribution, is a fundamental concept in statistics that represents a continuous probability distribution defined by a bell-shaped curve. It is symmetric around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

In the context of the paired t-test, checking for normal distribution of the differences between paired measurements is a crucial step. The reason behind this check is that the paired t-test assumes the differences are normally distributed. If this assumption is met, the test results are reliable; if not, they might lead us to incorrect conclusions. Statistical software or graphical methods such as Q-Q plots can help determine whether the differences are normally distributed.

When the differences are normally distributed, we expect most values to lie close to the mean difference with progressively fewer as we move away from the mean. If we find that the differences are skewed or have outliers, the data might not be normally distributed, and non-parametric tests like the Wilcoxon signed-rank test may be more suitable.

Statistically Significant Difference

After applying the appropriate statistical test, whether it be the paired t-test or the Wilcoxon signed-rank test, we reach the climax of our data analysis: determining if there is a statistically significant difference in MVD before and after tamoxifen treatment. Statistical significance is a term used to describe whether the results of a study are likely not due to random chance.

Researchers typically use a p-value to measure the probability that their results could have occurred under the null hypothesis (no effect). A common threshold for declaring statistical significance is a p-value of less than 0.05. If the p-value from our test is below this threshold, we conclude that the treatment has a significant effect on MVD - meaning there's a high likelihood that tamoxifen effectively reduces MVD in patients.

However, it's important to remember that 'statistically significant' does not necessarily mean 'clinically important'. Significance speaks only to the likelihood of the effect being real, not its magnitude or relevance in a practical context. Scientists must consider both statistical significance and the effect size when drawing conclusions from their studies.