Analyzing Hospital Data: Correlation Coefficients and Survival Rates

Abstract

Stat 5 Introduction to Statistics How, if at all, would the numbers in the table change if we instead computed the correlation coefficients between 30 Day Survival Percentage Rates and Number of Operations? (Note the change of order in the variables). The correlation coefficients would not change even if we interchanged the variables. The correlation values would remain the same as shown in the following table Let’s suppose that instead of using survival rate percentages we wanted to use survival rate decimals to analyze the hospital data. How would this change of units effect our correlation coefficients? Even if we use the survival rate values in decimal form, the correlation coefficients will remain the same. Using the scatterplots in figure 1, determine which correlation coefficient does a better job of summarizing the general direction (negative vs. positive) and strength (Weak to Strong) of the association between number of operations and 30-day survival percentage. Why did you make your decision and does your choice depend on whether or not Bristol is included in the analysis? The second scatter plot (Without Bristol) summarized the direction of graph better. It revealed that there was a strong positive correlation between the number of operations and 30-day survival percentage. I checked the general slope of the scatter plot and yes, my answer depends in the inclusion and the exclusion of Bristol. Given your knowledge of how the data was collected and the results shown in figure 1 and table 2 do you think it is reasonable to assume that making hospitals bigger causes higher survival rates in surgery? Why or why not? Making the hospital bigger does not increase the survival rates in surgery. The hospital should check on other factors such as the performance of the doctors, the number of staffs in the hospital, leadership aspects and management of the hospital. This is because Oxford Hospital performed the lowest number of operations but they had a survival rate of 81.1 % while Bristol performed 143 operations but they had the lowest survival rate of 71.3%. Statisticians often refer to robust statistics as statistics that are not unduly influenced by outliers. Using table 2 which statistic appears to be more robust to outliers Spearman or Pearson? Does this make sense given your knowledge of how these coefficients are calculated? The spearman Correlation is more robust to outliers than the Pearson correlations. The correlation increased more when the extreme outlier was removed while the Pearson correlation slightly decreased when the outlier was removed. Yes, this makes sense because the spearman correlation is calculated based on the ranks of the values, hence removing the outlier may not have a greater impact on the Spearman correlation. Looking at both of the scatter plots it appears as though there are 6 smaller hospitals (Bristol Included) which have lower survival rates than the 6 larger hospitals. Furthermore, there seems to be a possible pattern within the 6 smaller hospitals and no discernible pattern within the 6 larger hospitals. A possible explanation for this is that there is a diminishing return for number of operations performed which we can use as a proxy for hospital experience. Keeping this in mind suppose we want to try and measure the association between the number of operations and 30-Day Survival Percentage Rates within just the smaller hospitals. Below is the data subsetted for just the smallest hospitals. Table 3.0: Data from the hospitals with less than 200 operations Using this data compute, the • Pearson Correlation Coefficient is 0.6769 • Spearman Correlation Coefficient is 0.9429 Which correlation coefficient do you prefer and why? Pearson correlation since it measures the intensity of the linear association between the number of operations and the 30-Day survival rate. Question 2 Medical Research Gender Gap Often times when analyzing data, it is useful to define events and use probability theory to explore different aspects of the data. Following this framework lets define the events • A - the treatment is successful • B - drug 1 is given to a randomly chosen individual • C - the individual is female Using these events and the data above answer the following questions. Hint: The primary goal of this problem is to show how we can use the probability theory learned (Module 2) to increase our ability to interpret and explore data (Module 1). So, you should see some connections between the calculations. NOTE: you should round your answer to 4 decimal places, or use fractions What is the efficacy (proportion of successes) for Drug 1? What is the efficacy (proportion of successes) for Drug 2? Assuming that we did not have data by gender and only had these efficacies, which drug would you say is better? Drug 2 is better since it has a higher success rate when compared to drug 1 which has a very low success rate. (d) What is the efficacy (proportion of successes) for the women who received Drug 1? (e) What is...