An Introduction to Statistical Tests for the SAS Programmer Sara Beck, Fred Hutchinson Cancer Research Center, Seattle, WA

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1 ABSTRACT An Introduction to Statistical Tests for the SAS Programmer Sara Beck, Fred Hutchinson Cancer Research Center, Seattle, WA Often SAS Programmers find themselves in situations where performing a preliminary statistical analysis is beneficial. Basic statistical tests can help programmers better understand relationships between variables and notice when data aren't as expected. There is no shortage of resources for the statistician who uses SAS, but resources for the SAS programmer wanting to learn statistics are much more difficult to find. The aim of this paper is to help SAS programmers with no statistical training become comfortable coding and interpreting statistical tests in SAS. This paper discusses the following topics: An explanation of the definition of a statistical test and categorical, ordinal and interval types of variables. A brief discussion of commonly used statistical tests such as t-test, chi square, simple and multiple regression and ANOVA. Examples of how commonly used statistical tests are implemented in SAS, how to code dummy variables and what SAS options are useful. Tips for understanding statistical test output in SAS. Skills required to implement and interpret most basic tests are at an accessible level for most SAS programmers. Because SAS programmers are not statisticians, it may be difficult to know where to start, what to do, or how to interpret the results. INTRODUCTION Statistical tests can help SAS programmers better understand their data. Along with some intuition, statistical tests allow programmers to confirm relationships between variables and to catch mistakes in underlying assumptions about the data. Statistical tests can help programmers determine whether the data characteristics they see are statistically significant (i.e., not likely to be due to random variations). There are many resources for statisticians to better learn SAS, but unfortunately there are far fewer resources for SAS programmers who would like to learn basic statistics. Fortunately the skills required to implement statistical tests and understand their results are not out of grasp for SAS programmers. Understanding that most programmers are not statisticians, this paper aims to explain the most basic statistical concepts before explaining how to use statistical SAS procedures and interpret the results. It is not necessary to understand all of the SAS output from a statistical test to make some use of the results. This paper is geared at helping programmers learn the basics and helping them understand some SAS statistical test output. Most programmers are not statisticians, and obtaining in depth statistical details may not be the best use of their time when knowing the basics will suffice. Before even trying to run a statistical test in SAS, it is necessary to be aware of a few fundamental definitions. BASIC DEFINITIONS HYPOTHESIS TYPES A statistical test is a quantitative way to decide whether there is enough evidence to reasonably believe a conjecture to be true. Often statisticians think of these conjectures as complementing pairs of claims. These claims are usually referred to as the null hypothesis H 0, and the alternative hypothesis H a. Use of the term complementing is meant to imply that for any given situation the null and alternative hypotheses cannot both be true. These claims are not given the same weight; we do not reject the null hypothesis unless there is strong evidence against it. We can only have outcomes of reject H 0 in favor of H a, or do not reject H 0. A few examples of null and alternative hypothesis are the following: H 0: There is no taste difference between diet soda and full calorie soda. H a: There is a taste difference. H 0: Drug A and drug B are equally effective. H a: Drug B is more effective. H 0: The distribution of heights of adult women is normally distributed. H a: The distribution of heights of adult women is not normally distributed. 1

2 Note that the null and alternative hypotheses need not be exhaustive, as in the second example above. Here we implicitly assume that drug A cannot be more effective than drug B. To obtain correct results, it is important to determine whether the hypothesis tests are one or two-tailed. When the null and alternative hypotheses are of the form H 0: x 1= x 2, with H a: x 1> x 2 or H a: x 1< x 2, we call that a one-tailed test, and when the null hypothesis is of the form x 1 x 2, we call that a two tailed test. VARIABLE TYPES Before one is able to perform any statistical tests with variables, it is important to know the nature of variables involved. Each statistical test assumes its variables are of a certain type, and ones not of that type may simply not work. For example, with a variable for favorite color, there is no way to take the average, so a test comparing averages of favorite color would be nonsense. Categorical or nominal variables are ones such as favorite color, which have two or more categories and no way to order the values. Other examples of categorical variables include gender, blood type and favorite ice cream flavor. Ordinal variables can be ordered, but are similar to categorical variables in that there are clear categories. The relative distances or spacing between variables values is not uniform. For example, if we consider a the values of a survey variable: Strongly Disagree, Disagree, Neither Agree or Disagree, Agree, and Strongly Agree, we see that there is a clear order, but cannot speak to the true difference of Agree and Strongly Agree. Other examples of ordinal variables include place in competition or rankings minerals by hardness (Mohs scale of hardness.) Interval variables are similar to ordinal variables, except that values are measured in a way where their differences are meaningful. The place number of runners in a race is considered an ordinal scale, but if we consider the actual times of runners rather than their place, this would be an interval scale. Another example of an interval scale is the Celsius temperature scale. Some statistical tests assume the sample means are of a normal distribution (i.e.,the bell curve). If the sample size is sufficiently large, the central limit theorem guarantees the sample means are normally distributed. T-TESTS WHEN TO USE We can use t-tests in the following three situations; We want to test whether the mean is significantly different than a hypothesized value. We want to test whether means for two independent groups are significantly different. We want to test whether means for dependent or paired groups are significantly different. However, to use a t-test at all, we must have interval variables that are assumed normally distributed. HOW TO IMPLEMENT IN SAS To test whether the mean of one variable is significantly different than a hypothesized value, we can use the following SAS syntax: PROC TTEST DATA= datasetname H0=hypothesizedvalue; VAR variable_of_interest; If we omitted the H0=hypothesizedvalue option, SAS would use the default of H0=0 when running the t-test. In order to test whether the mean of two dependent groups are significantly different, we need to construct the SAS data set in such a way that we have two observations per subject. We use the following slightly different SAS syntax: PROC TTEST DATA= datasetname; PAIRED dependent_variablea*dependent_variableb; 2

3 Testing whether the means of two independent groups are different is the most complicated type of t-test. For this type of t-test, we need to create a classification variable or dummy variable. Class variables are 0/1 binary indicator variables. An example of a class variable might be gender, where gender=1 when the observation is male or gender=0 when the observation is female. Another example could be a vital status variable that equals 1 when a person is alive and is 0 when the person is dead. Once a class variable has been created, we use the following SAS syntax to perform the desired t-test. PROC TTEST DATA= datasetname; CLASS classvariable; VAR variable_of_interest; IMPORTANT RESULTS SAS will output other mean, standard deviation and confidence interval information pertaining to our t-test. For the purposed of accepting or rejecting the null hypothesis, we ll direct our attention towards the t-value, degrees of freedom and p-value. The p-value is your go-to value for this test. P-values indicate how likely the observation means were to occur from chance alone. The t-value itself is hard to interpret without use of degrees of freedom. In layman s terms, degrees of freedom is a value related to the number of observations and how variability is estimated. A negative or positive sign of the t-value indicates the observed mean is lower or higher than predicted respectively. SAS uses the t-value along with degrees of freedom to calculate the p-value. SAS will display the p-value result under the column heading: Pr > t, which is the correct p-value for a two-tailed test. For a one-tailed test, we simply divide the default SAS p- value by 2. Usually people will reject the null hypothesis with a p-value less than 0.05, though this line is arbitrary. EXAMPLE PROBLEM For the all example problems throughout this paper, we ll consider a fictitious dataset called vacation, containing data collected by a small international airline. Using PROC CONTENTS, we learn more about the data set and view the results below. Alphabetic List of Variables and Attributes # Variable Type Len Format Informat Label 9 Adult Num 8 All Adult Household 6 ContinentChange Num 8 Vacationed on a Foreign Continent 5 CountryChange Num 8 Vacationed in a Foreign Country 1 Family1D Num 8 Family1D 8 HouseholdSize Num 8 Number of Members in Household 7 Salary Num 8 Total Household Salary 4 Vcontinent Char 13 $13. $13. Continent Vacation Took Place 3 Vcost Num 8 Vacation Cost 2 Vlength Num 8 Length of Vacation 10 Season Char 6 $6. $6. Season Vacation Occurred Let s test to see if the mean length of vacation (VLength) is different between families of only adults and families of adults and children (Adult). We can set up the following pair of hypotheses: H 0: There is no difference between the mean length of vacation between adult families and adult-children families. H a: The mean vacation length for all adult families is greater than the mean vacation length for adult-children families. Assuming the data set vacation is in current SAS memory, we can write 3

4 PROC TTEST DATA= vacation; CLASS Adult; VAR Vlength; Running the above procedure for our fictitious data yields the following SAS results: The TTEST Procedure Variable: Vlength (Vlength) Adult N Mean Std Dev Std Err Minimum Maximum Diff (1-2) Adult Method Mean 95% CL Mean Std Dev 95% CL Std Dev Diff (1-2) Pooled Diff (1-2) Satterthwaite Method Variances DF t Value Pr > t Pooled Equal Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F Because we have a one-tailed test, and the SAS generated p-value is for a two-tailed test, we need to divide the calculated p-value by 2. Our p-value of indicates the mean vacation times between Adult only families and Adult-children families are not statistically different, so we continue to accept our null hypothesis. ONE-WAY ANOVA WHEN TO USE ANOVA can be thought of as a more generalized version of a t-test. If we compare only two means, both ANOVA and the t-test will yield the same results. Like t-tests, ANOVA requires normal interval variables. The aspect of ANOVA that is different from t-tests is the requirement of an independent categorical variable. We want to use oneway ANOVA when testing to see if the means of the interval dependent variable are different according to the independent categorical variable. HOW TO IMPLEMENT IN SAS There are two common ways to run ANOVA in SAS. A seemingly obvious way is PROC ANOVA, the other is PROC GLM, which has the added advantage of allowing with a few more SAS options. Below we see how we can use either procedure. PROC ANOVA has the following syntax: PROC ANOVA DATA= datasetname; CLASS ClassVariable; MODEL Response_Variable= ClassVariable; MEANS ClassVariable; 4

5 Alternatively, we can use the following syntax for PROC GLM.: PROC GML DATA= datasetname; CLASS ClassVariable; MODEL Response_Variable= ClassVariable; MEANS ClassVariable; There are many, many more options and ways the above SAS code can be elaborated, the above shows a simple way to run a one-way ANOVA in SAS. MEANS is not a required part of the procedure, but is nice to include as it will generate output for the means we re examining. IMPORTANT RESULTS SAS will output many statistical values after running either of the above statements. The most important values for programmers to understand from the SAS output are R 2, f-value, degrees of freedom, and the p-value. R 2 is the percentage of the variance from differences in the means from each category. The R 2 value helps quantify the relationship between the response variable and each of the class variable categories. A low R 2 indicates that there isn t much difference between groups. SAS calculates p-values from the f-value and degrees of freedom. A low p- value (usually p<0.05) is evidence against a null hypothesis. EXAMPLE PROBLEM Still examining the data set vacation, suppose we d like to test the following hypotheses about the average salary for families who took their vacations in different seasons. H 0: There is no difference between the mean salaries of families who vacationed in different seasons. H a: There is a difference between mean salaries of families who vacationed in different seasons. PROC ANOVA DATA= Vacation; CLASS Season; MODEL Salary= Season; MEANS Season; We obtain the following lengthy SAS output (on the next page) after running the above procedure: 5

6 The ANOVA Procedure Dependent Variable: Salary Total Household Salary Sum of Source DF Squares Mean Square F Value Pr > F Model Error E Corrected Total E12 R-Square Coeff Var Root MSE Salary Mean Source DF Anova SS Mean Square F Value Pr > F Vseason Dependent Variable: Salary Total Household Salary Sum of Source DF Squares Mean Square F Value Pr > F Model Error E Corrected Total E12 R-Square Coeff Var Root MSE Salary Mean Source DF Anova SS Mean Square F Value Pr > F Vseason Level of Salary Vseason N Mean Std Dev Fall Spring Summer Winter Because our p-value is (much) greater than.05, we accept our null hypothesis that there is no difference in the mean salary of each household with vacation season. CHI SQUARE GOODNESS OF FIT WHEN TO USE Programmers can use chi square goodness of fit to assess if frequencies of a categorical variable were likely to happen due to chance. Use of a chi square test is necessary whether proportions of a categorical variable are a hypothesized value. 6

7 HOW TO IMPLEMENT IN SAS To implement a chi square test, all we need to do is add the CHISQ option to a frequency procedure. To test whether proportions within a categorical value against a hypothesis, we use the following syntax: PROC FREQ DATA = datasetname; TABLES variable_of_interest / CHISQ TESTP=( ); The TESTP= option is necessary if the programmer would like to specify what proportions the chi square test is testing against. If the TESTP= option is omitted, SAS will assume the proportions within the category are equal. For a categorical variable with 4 possible values, the SAS default would be TESTP=( ). The proportions indicated in the the TESTP= option represent the null hypothesis. IMPORTANT RESULTS In addition to variable frequency results, SAS will output the following chi square specific results due to the CHISQ option : Chi-square, which is a number related to how much observations differ from the expected proportions. A large chi-square value comes from observed proportions quite different than what we expected, many observations in our data set or a combination of both. It is hard to interpret a chi-square value without considering degrees of freedom. Degrees of freedom (DF) are the number of categories in the analyzed variable minus one. P value, which indicates how likely the observed category proportions were to occur from chance alone based on our expected category proportions. A large chi-square value relative to the degrees of freedom indicates the observed category proportions are more drastically different than the expected proportions. When the p-value is less than.05 the null hypothesis is typically rejected, as a p-value less than.05 would correspond to less than 5% chance of rejecting the null hypothesis when it is indeed true. The lower a p-value is, the more significant the results. EXAMPLE PROBLEM Considering the dataset vacation, we d like to test the following hypotheses: H 0: 40% of vacations happened in summer, 25% happened in winter, 20% happened in spring and 15% happened in fall. H a: The percentages of vacations in each season are different than those listed. We can run the following SAS procedure to test these hypotheses, PROC FREQ DATA = vacation; TABLES Season / CHISQ TESTP=( ); And obtain the following SAS results: 7

8 The FREQ Procedure Season Vacation Occurred Test Cumulative Cumulative Vseason Frequency Percent Percent Frequency Percent Fall Spring Summer Winter Chi-Square Test for Specified Proportions Chi-Square DF 3 Pr > ChiSq <.0001 Sample Size = 199 Due to the small size of the p-value (<0.0001), we reject the null hypothesis in favor of the alternative hypothesis. LINEAR REGRESSION WHEN TO USE Simple linear regression is used when one wants to test how well a variable predicts another variable. Multiple linear regression is very similar, but allows one to test how well multiple variables predict a variable of interest. In order to use linear regression, we must be examining normally distributed interval variables. When using multiple linear regression, we additionally assume the predictor variables are independent. HOW TO IMPLEMENT IN SAS Running either or both simple and multiple linear regressions are very straightforward in SAS. Linear regression with one variable takes the following syntax: PROC REG datasetname; MODEL response = predictor / OPTIONS; Multiple linear regression has an approximately the same syntax as the simple linear regression. We simply add additional desired predictor variables in the model line, as so: PROC REG datasetname; MODEL response = predictora predictorb predictorc / OPTIONS; As with ANOVA, PROC GLM can also be used to run a linear regression. IMPORTANT RESULTS In addition to indicating whether or not there s a statistically significant linear relationship between variables, the SAS results will provide slope and intercept values for the best fit line through our data points. For a linear model, we hope the value of R-square is close to 1, as it is a measure of how well the predictor and response variables vary 8

9 together. SAS will list the intercept and slope of the best fit line, regardless of how well the best fit line models the data under a parameter estimates column. We still will use the p-value to tell whether our tests are statistically significant. EXAMPLE PROBLEM Still examining the data set vacation, we can test the following hypotheses with linear regression. H 0: There is no relationship between salary and amount spent on vacation. H a: There is a linear relationship between salary and amount spent on vacation. We can run the following SAS code to run the test, PROC REG vacation; MODEL Vcost = Salary; and ultimately obtain the following results. The REG Procedure Model: MODEL1 Dependent Variable: Vcost Cost of Vacation Number of Observations Read 199 Number of Observations Used 199 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model E10 <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > t Intercept Intercept Salary Total Household Salary E <.0001 Because the p-value is so small, we reject the null hypothesis that there is no relationship between salary and cost of vacation. The parameter estimate shows a positive relationship between salary and vacation cost. From the above results, we also know the best-fit line for our model is: vcost = 0.07(salary)

10 CONCLUSION Statistical tests may initially seem intimidating to SAS programmers with limited formal statistics background. Fortunately, SAS programmers can still benefit from statistical tests with only a basic statistical knowledge. Statistical fundamentals are within the aptitude range of programmers. Use of statistical tests can help programmers learn whether characteristics of their data are based purely on chance or are statistically significant, predict data values for future updates, discover data features, and ultimately help programmers maintain higher data quality standards. REFERENCES Evans, Micheal, and Jeffery Rosenthal. Probability and Statistics The Science of Uncertainty. 2nd ed. New York, NY: W.H. Freeman and Company, ,490-92, , Print. Barlow, R.J. Statistics A Guide to the Use of Statistical Methods in the Physical Sciences. New York, NY: Wiley, Print. Leeper, James. What Statistical Analysis Should I Use? UCLA: Academic Technology Services, Statistical Consulting Group. Web. Aug 2010 < Gerard, Dallal. "Degrees of Freedom." The Little Handbook of Statistical Practice. Web. 3 Sep < ACKNOWLEDGMENTS Special thanks to Nate Derby and Ben Ice for their help reviewing this paper. CONTACT INFORMATION Your comments and questions are valued and encouraged. Contact the author at: Sara Beck Fred Hutchinson Cancer Research Center SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are trademarks of their respective companies. 10

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