Question

SAS Descriptive Data Analysis Assignment

Purpose & Content of Analysis Paper

The purpose of this assignment is to provide a rough draft of a section of your analysis paper so that you can receive feedback on your methods. It should include the following:

1. Short description of the research question
2. Statement of the dataset to use to answer the question
3. Names of the variables used in your analysis. Use 2 main variables (e.g., weight status and depression) plus at least 3 other variables that could be associated with these main variables.
4. Descriptions of the variables used in the analysis
5. List of the SAS procedures you performed to provide descriptive statistics
6. Discussion of the results of those analyses

Answer 1

Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of virtually every quantitative analysis of data.

Descriptive statistics are typically distinguished from inferential statistics. With descriptive statistics you are simply describing what is or what the data shows. With inferential statistics, you are trying to reach conclusions that extend beyond the immediate data alone. For instance, we use inferential statistics to try to infer from the sample data what the population might think. Or, we use inferential statistics to make judgments of the probability that an observed difference between groups is a dependable one or one that might have happened by chance in this study. Thus, we use inferential statistics to make inferences from our data to more general conditions; we use descriptive statistics simply to describe what's going on in our data.

Descriptive Statistics are used to present quantitative descriptions in a manageable form. In a research study we may have lots of measures. Or we may measure a large number of people on any measure. Descriptive statistics help us to simplify large amounts of data in a sensible way. Each descriptive statistic reduces lots of data into a simpler summary. For instance, consider a simple number used to summarize how well a batter is performing in baseball, the batting average. This single number is simply the number of hits divided by the number of times at bat (reported to three significant digits). A batter who is hitting .333 is getting a hit one time in every three at bats. One batting .250 is hitting one time in four. The single number describes a large number of discrete events. Or, consider the scourge of many students, the Grade Point Average (GPA). This single number describes the general performance of a student across a potentially wide range of course experiences.

Every time you try to describe a large set of observations with a single indicator you run the risk of distorting the original data or losing important detail. The batting average doesn't tell you whether the batter is hitting home runs or singles. It doesn't tell whether she's been in a slump or on a streak. The GPA doesn't tell you whether the student was in difficult courses or easy ones, or whether they were courses in their major field or in other disciplines. Even given these limitations, descriptive statistics provide a powerful summary that may enable comparisons across people or other units.

Univariate Analysis

Univariate analysis involves the examination across cases of one variable at a time. There are three major characteristics of a single variable that we tend to look at:

the distribution

the central tendency

the dispersion

In most situations, we would describe all three of these characteristics for each of the variables in our study.

The Distribution. The distribution is a summary of the frequency of individual values or ranges of values for a variable. The simplest distribution would list every value of a variable and the number of persons who had each value. For instance, a typical way to describe the distribution of college students is by year in college, listing the number or percent of students at each of the four years. Or, we describe gender by listing the number or percent of males and females. In these cases, the variable has few enough values that we can list each one and summarize how many sample cases had the value. But what do we do for a variable like income or GPA? With these variables there can be a large number of possible values, with relatively few people having each one. In this case, we group the raw scores into categories according to ranges of values. For instance, we might look at GPA according to the letter grade ranges. Or, we might group income into four or five ranges of income values.

One of the most common ways to describe a single variable is with a frequency distribution. Depending on the particular variable, all of the data values may be represented, or you may group the values into categories first (e.g., with age, price, or temperature variables, it would usually not be sensible to determine the frequencies for each value. Rather, the value are grouped into ranges and the frequencies determined.). Frequency distributions can be depicted in two ways, as a table or as a graph.

Distributions may also be displayed using percentages. For example, you could use percentages to describe the:

percentage of people in different income levels

percentage of people in different age ranges

percentage of people in different ranges of standardized test scores

Central Tendency. The central tendency of a distribution is an estimate of the "center" of a distribution of values. There are three major types of estimates of central tendency:

Mean

Median

Mode

The Mean or average is probably the most commonly used method of describing central tendency. To compute the mean all you do is add up all the values and divide by the number of values. For example, the mean or average quiz score is determined by summing all the scores and dividing by the number of students taking the exam. For example, consider the test score values:

15, 20, 21, 20, 36, 15, 25, 15

The sum of these 8 values is 167, so the mean is 167/8 = 20.875.

The Median is the score found at the exact middle of the set of values. One way to compute the median is to list all scores in numerical order, and then locate the score in the center of the sample. For example, if there are 500 scores in the list, score #250 would be the median. If we order the 8 scores shown above, we would get:

15,15,15,20,20,21,25,36

There are 8 scores and score #4 and #5 represent the halfway point. Since both of these scores are 20, the median is 20. If the two middle scores had different values, you would have to interpolate to determine the median.

The mode is the most frequently occurring value in the set of scores. To determine the mode, you might again order the scores as shown above, and then count each one. The most frequently occurring value is the mode. In our example, the value 15 occurs three times and is the model. In some distributions there is more than one modal value. For instance, in a bimodal distribution there are two values that occur most frequently.

Notice that for the same set of 8 scores we got three different values -- 20.875, 20, and 15 -- for the mean, median and mode respectively. If the distribution is truly normal (i.e., bell-shaped), the mean, median and mode are all equal to each other.

Dispersion. Dispersion refers to the spread of the values around the central tendency. There are two common measures of dispersion, the range and the standard deviation. The range is simply the highest value minus the lowest value. In our example distribution, the high value is 36 and the low is 15, so the range is 36 - 15 = 21.

The Standard Deviation is a more accurate and detailed estimate of dispersion because an outlier can greatly exaggerate the range (as was true in this example where the single outlier value of 36 stands apart from the rest of the values. The Standard Deviation shows the relation that set of scores has to the mean of the sample. Again lets take the set of scores:

15,20,21,20,36,15,25,15

to compute the standard deviation, we first find the distance between each value and the mean. We know from above that the mean is 20.875. So, the differences from the mean are:

15 - 20.875 = -5.875
20 - 20.875 = -0.875
21 - 20.875 = +0.125
20 - 20.875 = -0.875
36 - 20.875 = 15.125
15 - 20.875 = -5.875
25 - 20.875 = +4.125
15 - 20.875 = -5.875

Notice that values that are below the mean have negative discrepancies and values above it have positive ones. Next, we square each discrepancy:

-5.875 * -5.875 = 34.515625
-0.875 * -0.875 = 0.765625
+0.125 * +0.125 = 0.015625
-0.875 * -0.875 = 0.765625
15.125 * 15.125 = 228.765625
-5.875 * -5.875 = 34.515625
+4.125 * +4.125 = 17.015625
-5.875 * -5.875 = 34.515625

Now, we take these "squares" and sum them to get the Sum of Squares (SS) value. Here, the sum is 350.875. Next, we divide this sum by the number of scores minus 1. Here, the result is 350.875 / 7 = 50.125. This value is known as the variance. To get the standard deviation, we take the square root of the variance (remember that we squared the deviations earlier). This would be SQRT(50.125) = 7.079901129253.

Although this computation may seem convoluted, it's actually quite simple. To see this, consider the formula for the standard deviation:

In the top part of the ratio, the numerator, we see that each score has the the mean subtracted from it, the difference is squared, and the squares are summed. In the bottom part, we take the number of scores minus 1. The ratio is the variance and the square root is the standard deviation. In English, we can describe the standard deviation as:

the square root of the sum of the squared deviations from the mean divided by the number of scores minus one

Although we can calculate these univariate statistics by hand, it gets quite tedious when you have more than a few values and variables. Every statistics program is capable of calculating them easily for you. For instance, I put the eight scores into SPSS and got the following table as a result:

N8

Mean20.8750

Median20.0000

Mode15.00

Std. Deviation7.0799

Variance50.1250

Range21.00

which confirms the calculations I did by hand above.

The standard deviation allows us to reach some conclusions about specific scores in our distribution. Assuming that the distribution of scores is normal or bell-shaped (or close to it!), the following conclusions can be reached:

approximately 68% of the scores in the sample fall within one standard deviation of the mean

approximately 95% of the scores in the sample fall within two standard deviations of the mean

approximately 99% of the scores in the sample fall within three standard deviations of the mean

For instance, since the mean in our example is 20.875 and the standard deviation is 7.0799, we can from the above statement estimate that approximately 95% of the scores will fall in the range of 20.875-(2*7.0799) to 20.875+(2*7.0799) or between 6.7152 and 35.0348. This kind of information is a critical stepping stone to enabling us to compare the performance of an individual on one variable with their performance on another, even when the variables are measured on entirely different scales.

for sas procedures

DESCRIPTIVE STATISTICS SAS LEARNING MODULES

This module illustrates how to obtain basic descriptive statistics using SAS. We illustrate this using a data file about 26 automobiles with their make, price, mpg, repair record, and whether the car was foreign or domestic. The data file is illustrated below.

MAKE PRICE MPG REP78 FOREIGN AMC 4099 22 3 0 AMC 4749 17 3 0 AMC 3799 22 3 0 Audi 9690 17 5 1 Audi 6295 23 3 1 BMW 9735 25 4 1 Buick 4816 20 3 0 Buick 7827 15 4 0 Buick 5788 18 3 0 Buick 4453 26 3 0 Buick 5189 20 3 0 Buick 10372 16 3 0 Buick 4082 19 3 0 Cad. 11385 14 3 0 Cad. 14500 14 2 0 Cad. 15906 21 3 0 Chev. 3299 29 3 0 Chev. 5705 16 4 0 Chev. 4504 22 3 0 Chev. 5104 22 2 0 Chev. 3667 24 2 0 Chev. 3955 19 3 0 Datsun 6229 23 4 1 Datsun 4589 35 5 1 Datsun 5079 24 4 1 Datsun 8129 21 4 1

The program below reads the data and creates a temporary data file called auto. The descriptive statistics shown in this module are all performed on this data file called auto.

DATA auto ; input MAKE $ PRICE MPG REP78 FOREIGN ; DATALINES; AMC 4099 22 3 0 AMC 4749 17 3 0 AMC 3799 22 3 0 Audi 9690 17 5 1 Audi 6295 23 3 1 BMW 9735 25 4 1 Buick 4816 20 3 0 Buick 7827 15 4 0 Buick 5788 18 3 0 Buick 4453 26 3 0 Buick 5189 20 3 0 Buick 10372 16 3 0 Buick 4082 19 3 0 Cad. 11385 14 3 0 Cad. 14500 14 2 0 Cad. 15906 21 3 0 Chev. 3299 29 3 0 Chev. 5705 16 4 0 Chev. 4504 22 3 0 Chev. 5104 22 2 0 Chev. 3667 24 2 0 Chev. 3955 19 3 0 Datsun 6229 23 4 1 Datsun 4589 35 5 1 Datsun 5079 24 4 1 Datsun 8129 21 4 1 ; RUN; PROC PRINT DATA=auto(obs=10); RUN;

The output of the proc print is shown below. You can compare the program above to the output below.

OBS MAKE PRICE MPG REP78 FOREIGN 1 AMC 4099 22 3 0 2 AMC 4749 17 3 0 3 AMC 3799 22 3 0 4 Audi 9690 17 5 1 5 Audi 6295 23 3 1 6 BMW 9735 25 4 1 7 Buick 4816 20 3 0 8 Buick 7827 15 4 0 9 Buick 5788 18 3 0 10 Buick 4453 26 3 0

2. Using proc freq for frequencies

We can use proc freq to produce frequency tables. Below, we use it to make frequency tables for make, rep78 and foreign.

PROC FREQ DATA=auto; TABLES make ; RUN; PROC FREQ DATA=auto; TABLES rep78 ; RUN; PROC FREQ DATA=auto; TABLES foreign ; RUN;

Here is the output produced by the proc freq statements above.

Cumulative Cumulative MAKE Frequency Percent Frequency Percent ---------------------------------------------------- AMC 3 11.5 3 11.5 Audi 2 7.7 5 19.2 BMW 1 3.8 6 23.1 Buick 7 26.9 13 50.0 Cad. 3 11.5 16 61.5 Chev. 6 23.1 22 84.6 Datsun 4 15.4 26 100.0 Cumulative Cumulative REP78 Frequency Percent Frequency Percent --------------------------------------------------- 2 3 11.5 3 11.5 3 15 57.7 18 69.2 4 6 23.1 24 92.3 5 2 7.7 26 100.0 Cumulative Cumulative FOREIGN Frequency Percent Frequency Percent ----------------------------------------------------- 0 19 73.1 19 73.1 1 7 26.9 26 100.0

Instead of having three separate proc freqs, we could have done this all in one proc freq step as illustrated below. The output will be the same as shown above.

PROC FREQ DATA=auto; TABLES make rep78 foreign ; RUN;

Let’s use proc freq to look at a cross tabulation of the repair history of the cars (rep78) for foreign and domestic cars (foreign). The proc freqstatements for this are shown below. Note the asterisk (*) between the variables rep78 and foreign on the tables statement.

PROC FREQ DATA=auto; TABLES rep78*foreign ; RUN;

This is the output produced.

TABLE OF REP78 BY FOREIGN REP78 FOREIGN Frequency| Percent | Row Pct | Col Pct | 0| 1| Total ---------+--------+--------+ 2 | 3 | 0 | 3 | 11.54 | 0.00 | 11.54 | 100.00 | 0.00 | | 15.79 | 0.00 | ---------+--------+--------+ 3 | 14 | 1 | 15 | 53.85 | 3.85 | 57.69 | 93.33 | 6.67 | | 73.68 | 14.29 | ---------+--------+--------+ 4 | 2 | 4 | 6 | 7.69 | 15.38 | 23.08 | 33.33 | 66.67 | | 10.53 | 57.14 | ---------+--------+--------+ 5 | 0 | 2 | 2 | 0.00 | 7.69 | 7.69 | 0.00 | 100.00 | | 0.00 | 28.57 | ---------+--------+--------+ Total 19 7 26 73.08 26.92 100.00

We can show just the cell percentages to make the table easier to read by using the norow, nocol and nofreq options on the tablesstatement to suppress the printing of the row percentages, column percentages and frequencies (leaving just the cell percentages). Note that the options come after the forward slash ( / ) on the tables statement.

PROC FREQ DATA=auto; TABLES rep78*foreign / NOROW NOCOL NOFREQ ; RUN;

The output is shown below.

TABLE OF REP78 BY FOREIGN REP78 FOREIGN Percent | 0| 1| Total --------+--------+--------+ 2 | 11.54 | 0.00 | 11.54 --------+--------+--------+ 3 | 53.85 | 3.85 | 57.69 --------+--------+--------+ 4 | 7.69 | 15.38 | 23.08 --------+--------+--------+ 5 | 0.00 | 7.69 | 7.69 --------+--------+--------+ Total 19 7 26 73.08 26.92 100.00

The order of the options does not matter. We would have gotten the same output had we written the command like this.

PROC FREQ DATA=auto; TABLES rep78*foreign / NOFREQ NOROW NOCOL ; RUN;

3. Using proc means for summary statistics

Proc means can be used to produce summary statistics. Below, proc means is used to get descriptive statistics for the variable mpg.

PROC MEANS DATA=auto; VAR mpg; RUN;

The results of the proc means are shown below.

Analysis Variable : MPG N Mean Std Dev Minimum Maximum ---------------------------------------------------------- 26 20.9230769 4.7575042 14.0000000 35.0000000 ----------------------------------------------------------

Suppose we would like to get the summary statistics separately for foreign and domestic cars (indicated by the variable foreign). We can use the class statement (shown below) to get separate results for the different values of foreign.

PROC MEANS DATA=auto; CLASS foreign ; VAR mpg; RUN;

As you see below, the results of  proc means are presented separately for the seven foreign cars (when foreign equals 1) and the 19 domestic cars (when foreign equals 0).

Analysis Variable : MPG FOREIGN N Obs N Mean Std Dev Minimum Maximum ------------------------------------------------------------- 0 19 19 19.78 4.0356598 14.0000 29.00 1 7 7 24.00 5.5075705 17.0000 35.00 --------------------------------------------------------------

4. Using proc univariate for detailed summary statistics

You can use proc univariate to get more detailed summary statistics, as shown below.

PROC UNIVARIATE DATA=auto; VAR mpg; RUN;

And here are the results of proc univariate.

Univariate Procedure Variable=MPG Moments N 26 Sum Wgts 26 Mean 20.92308 Sum 544 Std Dev 4.757504 Variance 22.63385 Skewness 0.935473 Kurtosis 1.7927 USS 11948 CSS 565.8462 CV 22.73807 Std Mean 0.933023 T:Mean=0 22.42503 Pr>|T| 0.0001 Num ^= 0 26 Num > 0 26 M(Sign) 13 Pr>=|M| 0.0001 Sgn Rank 175.5 Pr>=|S| 0.0001 Quantiles(Def=5) 100% Max 35 99% 35 75% Q3 23 95% 29 50% Med 21 90% 26 25% Q1 17 10% 15 0% Min 14 5% 14 1% 14 Range 21 Q3-Q1 6 Mode 22 Extremes Lowest Obs Highest Obs 14( 15) 24( 25) 14( 14) 25( 6) 15( 8) 26( 10) 16( 18) 29( 17) 16( 12) 35( 24)

We can use the class statement to obtain separate univariate results for foreign and domestic cars.

PROC UNIVARIATE DATA=auto; CLASS foreign; VAR mpg; RUN;

As you see in the output below, you get a complete set of output for the case when foreign equals 0 and then another set of output when foreign equals 1.

The UNIVARIATE Procedure Variable: MPG FOREIGN = 0 Moments N 19 Sum Weights 19 Mean 19.7894737 Sum Observations 376 Std Deviation 4.03565976 Variance 16.2865497 Skewness 0.477379 Kurtosis 0.04119835 Uncorrected SS 7734 Corrected SS 293.157895 Coeff Variation 20.3929616 Std Error Mean 0.92584385 Basic Statistical Measures Location Variability Mean 19.78947 Std Deviation 4.03566 Median 20.00000 Variance 16.28655 Mode 22.00000 Range 15.00000 Interquartile Range 6.00000 Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t t 21.37453 Pr > |t| <.0001 Sign M 9.5 Pr >= |M| <.0001 Signed Rank S 95 Pr >= |S| <.0001 Quantiles (Definition 5) Quantile Estimate 100% Max 29 99% 29 95% 29 90% 26 75% Q3 22 50% Median 20 25% Q1 16 10% 14 5% 14 1% 14 0% Min 14 Variable: MPG FOREIGN = 0 Extreme Observations ----Lowest---- ----Highest--- Value Obs Value Obs 14 15 22 19 14 14 22 20 15 8 24 21 16 18 26 10 16 12 29 17 Variable: MPG FOREIGN = 1 Moments N 7 Sum Weights 7 Mean 24 Sum Observations 168 Std Deviation 5.50757055 Variance 30.3333333 Skewness 1.34081176 Kurtosis 3.28605241 Uncorrected SS 4214 Corrected SS 182 Coeff Variation 22.9482106 Std Error Mean 2.081666 Basic Statistical Measures Location Variability Mean 24.00000 Std Deviation 5.50757 Median 23.00000 Variance 30.33333 Mode 23.00000 Range 18.00000 Interquartile Range 4.00000 Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t t 11.52923 Pr > |t| <.0001 Sign M 3.5 Pr >= |M| 0.0156 Signed Rank S 14 Pr >= |S| 0.0156 Quantiles (Definition 5) Quantile Estimate 100% Max 35 99% 35 95% 35 90% 35 75% Q3 25 50% Median 23 25% Q1 21 10% 17 5% 17 1% 17 0% Min 17 Variable: MPG FOREIGN = 1 Extreme Observations ----Lowest---- ----Highest--- Value Obs Value Obs 17 4 23 5 21 26 23 23 23 23 24 25 23 5 25 6 24 25 35 24

5. Problems to look out for

If you make a crosstab with proc freq and one of the variables has large number of values (say 10 or more) the crosstab table could be very hard to read. In such cases, try using the list option on the tables statement.

TABLES rep78*foreign / LIST ;

When using the by statement in proc univariate, if you choose a by variable with a large number of values (say 5, 10, or more) it will produce a very large amount of output. In such cases, you may try to use proc means with a class statement instead of proc univariate.