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The following data are a sample of survival times (days from diagnosis) for patients suffering from chronic leukemia of a certain type (Statistical Methodology for Survival Time Studies [Bethesda, MD: National Cancer Institute, 1986\(]\) ): $$ \begin{array}{rrrrrrrr} 7 & 47 & 58 & 74 & 177 & 232 & 273 & 285 \\ 317 & 429 & 440 & 445 & 455 & 468 & 495 & 497 \\ 532 & 571 & 579 & 581 & 650 & 702 & 715 & 779 \\ 881 & 900 & 930 & 968 & 1077 & 1109 & 1314 & 1334 \\ 1367 & 1534 & 1712 & 1784 & 1877 & 1886 & 2045 & 2056 \\ 2260 & 2429 & 2509 & & & & & \end{array} $$ a. Construct a relative frequency distribution for this data set, and draw the corresponding histogram. b. Would you describe this histogram as having a positive or a negative skew? c. Would you recommend transforming the data? Explain.

Step by step solution

01

Organize data in ascending order

This will allow for easy identification of the range and distribution of data points. It is important to note that our smallest data point is 7 and the largest is 2509.

02

Construct a relative frequency distribution

The data range, which is 2502 (2509-7), can be divided by the desired number of intervals (usually under 10 for clarity). The class width (range of each interval) can be approximated to a convenient number. For example, if we choose 8 intervals, our class width rounds to 315. To construct the frequency distribution, count how many data points fall inside each interval (e.g., 0-315, 316-630, and so on). The relative frequency is calculated by dividing the count for each interval by the total number of data points.

03

Draw the histogram

Each interval is represented by a rectangular bar, whose height corresponds to the relative frequency of that interval. The bars should be adjacent to each other, with no gaps between them.

04

Analyze the skewness

Analyze the shape of the histogram. If the right (upper-end) tail is longer, the histogram is positively skewed, otherwise it's negatively skewed.

05

Comment on possible data transformation

If the histogram is heavily skewed – either positively or negatively – data transformation may be considered to normalize the distribution. The recommendation will be made based on skewness observed in step 4.

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Key Concepts

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

Relative Frequency Distribution

A relative frequency distribution is a valuable tool to understand how data is spread across different intervals. It shows the proportion of data points that lie within each interval, which helps in comparing the magnitude of different parts of the dataset.
To construct a relative frequency distribution:

• First, order the data from smallest to largest. This helps to determine the range, which in our exercise was from 7 to 2509.
• Next, decide on a suitable number of intervals, often aiming for fewer than 10 for clarity. This can be calculated by dividing the data range by the number of intervals to obtain a class width. For instance, with an 8-interval choice, our class width was 315.
• Count the number of data points within each interval, then divide this count by the total number of data points to get the relative frequency.

This method provides insight into how data points are distributed across different ranges, adding a relative perspective that is often more informative than just raw frequency counts.

Histogram

A histogram is a graphical representation of data distribution. It is specifically used to understand the shape and spread of continuous data.
Here's how a histogram is created using our relative frequency distribution:

• Each interval from the relative frequency distribution is represented by a bar on the histogram.
• The height of each bar corresponds to the relative frequency of the interval it represents.
• These bars are placed next to each other with no gaps, showing a continuous data distribution.

By visually analyzing the histogram, one can quickly identify the data's distribution pattern and any noticeable trends or outliers. It serves as a practical visual aid for interpreting complex datasets, such as survival times in our example.

Skewness

Skewness describes the degree of asymmetry of a dataset's distribution. This is crucial in survival analysis as it affects the interpretation of survival times.
A histogram can help determine skewness through its shape:

• If the longer tail of the histogram is on the right-hand side, the data is positively skewed, indicating more data values are clustered at the lower end.
• If the longer tail is on the left, it's negatively skewed, showing a clustering of values at the higher end.

For the given dataset, observing the histogram will help to see whether the data tends to skew towards shorter or longer survival times. Identifying skewness is essential as it might influence further statistical analyses and the need for data transformations.

Data Transformation

Data transformation involves changing the data using a mathematical function to make it more suitable for analysis. This is often done to correct skewness.
Transformations can help modify skewed distributions to resemble a normal distribution more closely:

• Common methods include taking the logarithm, square root, or inverse of the data values.
• This can stabilize variance and make patterns in data more observable and understandable.

In survival analysis, using such transformations can help in accurately modeling and interpreting survival times, ensuring that statistical assumptions meet inferences drawn from the data. Whether a transformation is needed typically depends on the degree of skewness observed in the data's histogram.