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HLTH 653 Lecture 2

HLTH 653 Lecture 2. Raul Cruz-Cano Spring 2013. Statistical analysis procedures. Proc univariate Proc t test Proc corr Proc reg. Proc Univariate. The UNIVARIATE procedure provides data summarization on the distribution of numeric variables. PROC UNIVARIATE < option(s) >;

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HLTH 653 Lecture 2

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  1. HLTH 653 Lecture 2 Raul Cruz-Cano Spring 2013

  2. Statistical analysis procedures • Proc univariate • Proc t test • Proc corr • Proc reg

  3. Proc Univariate • The UNIVARIATE procedure provides data summarization on the distribution of numeric variables. PROC UNIVARIATE<option(s)>; Varvariable-1variable-n; Run; Options: PLOTS : create low-resolution stem-and-leaf, box, and normal probability plots NORMAL: Request tests for normality

  4. data blood; INFILE'C:\blood.txt'; INPUTsubjectID $ gender $ bloodtype $ age_group$ RBC WBC cholesterol; run; proc univariate data =blood ; varcholesterol; run;

  5. OUTPUT (1) The UNIVARIATE Procedure Variable: cholesterol Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947

  6. Moments - Moments are statistical summaries of a distribution OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 Sum Weights - A numeric variable can be specified as a weight variable to weight the values of the analysis variable. The default weight variable is defined to be 1 for each observation. This field is the sum of observation values for the weight variable N - This is the number of valid observations for the variable. The total number of observations is the sum of N and the number of missing values.

  7. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 Sum Observations - This is the sum of observation values. In case that a weight variable is specified, this field will be the weighted sum. The mean for the variable is the sum of observations divided by the sum of weights.

  8. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 Std Deviation - Standard deviation is the square root of the variance. It measures the spread of a set of observations. The larger the standard deviation is, the more spread out the observations are.

  9. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 Variance - The variance is a measure of variability. It is the sum of the squared distances of data value from the mean divided by N-1. We don't generally use variance as an index of spread because it is in squared units. Instead, we use standard deviation.

  10. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 Skewness - Skewness measures the degree and direction of asymmetry. A symmetric distribution such as a normal distribution has a skewness of 0, and a distribution that is skewed to the left, e.g. when the mean is less than the median, has a negative skewness.

  11. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 (1)Kurtosis - Kurtosis is a measure of the heaviness of the tails of a distribution. In SAS, a normal distribution has kurtosis 0. (2) Extremely nonnormal distributions may have high positive or negative kurtosis values, while nearly normal distributions will have kurtosis values close to 0. (3) Kurtosis is positive if the tails are "heavier" than for a normal distribution and negative if the tails are "lighter" than for a normal distribution.

  12. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 Uncorrected Sum of Square Distances from the Mean - This is the sum of squared data values. Corrected SS - This is the sum of squared distance of data values from the mean. This number divided by the number of observations minus one gives the variance.

  13. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 (1)Coeff Variation - The coefficient of variation is another way of measuring variability. (2)It is a unitless measure. (3)It is defined as the ratio of the standard deviation to the mean and is generally expressed as a percentage. (4) It is useful for comparing variation between different variables.

  14. OUTPUT (1) Moments N 795 Sum Weights 795 Mean 201.43522 Sum Observations 160141 Std Deviation 49.8867157 Variance 2488.6844 Skewness -0.0014449 Kurtosis -0.0706044 Uncorrected SS 34234053 Corrected SS 1976015.41 Coeff Variation 24.7656371 Std Error Mean 1.76929947 (1)Std Error Mean - This is the estimated standard deviation of the sample mean. (2)It is estimated as the standard deviation of the sample divided by the square root of sample size. (3)This provides a measure of the variability of the sample mean.

  15. OUTPUT (2) Location Variability Mean 201.4352 Std Deviation 49.88672 Median 202.0000 Variance 2489 Mode 208.0000 Range 314.00000 Interquartile Range 71.00000 NOTE: The mode displayed is the smallest of 2 modes with a count of 12. Median - The median is a measure of central tendency. It is the middle number when the values are arranged in ascending (or descending) order. It is less sensitive than the mean to extreme observations. Mode - The mode is another measure of central tendency. It is the value that occurs most frequently in the variable.

  16. OUTPUT (3) Location Variability Mean 201.4352 Std Deviation 49.88672 Median 202.0000 Variance 2489 Mode 208.0000 Range 314.00000 Interquartile Range 71.00000 NOTE: The mode displayed is the smallest of 2 modes with a count of 12. Range - The range is a measure of the spread of a variable. It is equal to the difference between the largest and the smallest observations. It is easy to compute and easy to understand. Interquartile Range - The interquartile range is the difference between the upper (75% Q) and the lower quartiles (25% Q). It measures the spread of a data set. It is robust to extreme observations.

  17. OUTPUT (3) Tests for Location: Mu0=0 Test -Statistic- -----p Value------ Student's t t 113.8503 Pr > |t| <.0001 Sign M 397.5 Pr >= |M| <.0001 Signed Rank S 158205 Pr >= |S| <.0001

  18. OUTPUT (3) Student's t t 113.8503 Pr > |t| <.0001 Sign M 397.5 Pr >= |M| <.0001 Signed Rank S 158205 Pr >= |S| <.0001 (1) Sign - The sign test is a simple nonparametric procedure to test the null hypothesis regarding the population median. (2) It is used when we have a small sample from a nonnormal distribution. (3)The statistic M is defined to be M=(N+-N-)/2 where N+ is the number of values that are greater than Mu0 and N- is the number of values that are less than Mu0. Values equal to Mu0 are discarded. (4)Under the hypothesis that the population median is equal to Mu0, the sign test calculates the p-value for M using a binomial distribution. (5)The interpretation of the p-value is the same as for t-test. In our example the M-statistic is 398 and the p-value is less than 0.0001. We conclude that the median of variable is significantly different from zero.

  19. OUTPUT (3) Student's t t 113.8503 Pr > |t| <.0001 Sign M 397.5 Pr >= |M| <.0001 Signed Rank S 158205 Pr >= |S| <.0001 (1) Signed Rank - The signed rank test is also known as the Wilcoxon test. It is used to test the null hypothesis that the population median equals Mu0. (2) It assumes that the distribution of the population is symmetric. (3)The Wilcoxon signed rank test statistic is computed based on the rank sum and the numbers of observations that are either above or below the median. (4) The interpretation of the p-value is the same as for the t-test. In our example, the S-statistic is 158205 and the p-value is less than 0.0001. We therefore conclude that the median of the variable is significantly different from zero.

  20. OUTPUT (4) Quantiles (Definition 5) Quantile Estimate 100% Max 331 99% 318 95% 282 90% 267 75% Q3 236 50% Median 202 25% Q1 165 10% 138 5% 123 1% 94 0% Min 17 95% - Ninety-five percent of all values of the variable are equal to or less than this value.

  21. OUTPUT (5) Extreme Observations ----Lowest---- ----Highest--- Value Obs Value Obs 17 829 323 828 36 492 328 203 56 133 328 375 65 841 328 541 69 79 331 191 Missing Values -----Percent Of----- Missing Missing Value Count All ObsObs . 205 20.50 100.00 Extreme Observations - This is a list of the five lowest and five highest values of the variable

  22. Student's t-test • Independent One-Sample t-test • This equation is used to compare one sample mean to a specific value μ0. • Where s is the grand standard deviation of the sample. N is the sample size. The degrees of freedom used in this test is N-1.

  23. Student's t-test • Dependent t-test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". • For this equation, the differences between all pairs must be calculated. The pairs are either one person's pretest and posttest scores or one person in a group matched to another person in another group. The average (XD) and standard deviation (sD) of those differences are used in the equation. The constant μ0 is non-zero if you want to test whether the average of the difference is significantly different than μ0. The degree of freedom used is N-1.

  24. PROC TTEST The following statements are available in PROC TTEST. PROC TTEST < options > ; CLASS variable ; PAIRED variables ; BY variables ; VAR variables ; CLASS: CLASS statement giving the name of the classification (or grouping) variable must accompany the PROC TTEST statement in the two independent sample cases (TWO SAMPLE T TEST). The class variable must have two, and only two, levels.

  25. Paired Statements • PAIRED: the PAIRED statement identifies the variables to be compared in paired t test • You can use one or more variables in the PairLists. • Variables or lists of variables are separated by an asterisk (*) or a colon (:). • The asterisk (*) requests comparisons between each variable on the left with each variable on the right. • Use the PAIRED statement only for paired comparisons. • The CLASS and VAR statements cannot be used with the PAIRED statement.

  26. PROC TTEST OPTIONS : ALPHA=p specifies that confidence intervals are to be 100(1-p)% confidence intervals, where 0<p<1. By default, PROC TTEST uses ALPHA=0.05. If p is 0 or less, or 1 or more, an error message is printed. H0=m requests tests against m instead of 0 in all three situations (one-sample, two-sample, and paired observation t tests). By default, PROC TTEST uses H0=0. DATA=SAS-data-set names the SAS data set for the procedure to use

  27. *One sample ttest*; Proc ttest data =blood H0=200; varcholesterol; run;

  28. One sample t test Output The TTEST Procedure Variable: cholesterol N Mean Std Dev Std Err Minimum Maximum 795 201.4 49.8867 1.7693 17.0000 331.0 Mean 95% CL Mean Std Dev 95% CL Std Dev 201.4 198.0 204.9 49.8867 47.5493 52.4676 DF t Value Pr > |t| 794 0.81 0.4175 95%CL Std Dev is 95% confidence interval for the standard deviation. 95%CL Mean is 95% confidence interval for the mean.

  29. One sample t test Output It is the probability of observing a greater absolute value of t under the null hypothesis.  Variable: cholesterol N Mean Std Dev Std Err Minimum Maximum 795 201.4 49.8867 1.7693 17.0000 331.0 Mean 95% CL Mean Std Dev 95% CL Std Dev 201.4 198.0 204.9 49.8867 47.5493 52.4676 DF t Value Pr > |t| 794 0.81 0.4175 T value is the t-statistic.  It is the ratio of the difference between the sample mean and the given number to the standard error of the mean.  DF - The degrees of freedom for the t-test is simply the number of valid observations minus 1.  We loose one degree of freedom because we have estimated the mean from the sample.  We have used some of the information from the data to estimate the mean; therefore, it is not available to use for the test and the degrees of freedom accounts for this

  30. title 'Paired Comparison'; datapressure; input SBPbeforeSBPafter @@; diff_BP=SBPafter-SBPbefore; datalines; 120 128 124 131 130 131 118 127 140 132 128 125 140 141 135 137 126 118 130 132 126 129 127 135 ; run; proc ttestdata=pressure; paired SBPbefore*SBPafter; run;

  31. Paired t test Output The TTEST Procedure Difference: SBPbefore - SBPafter N Mean Std Dev Std Err Minimum Maximum 12 -1.8333 5.8284 1.6825 -9.0000 8.0000 Mean 95% CL Mean Std Dev 95% CL Std Dev -1.8333 -5.5365 1.8698 5.8284 4.1288 9.8958 DF t Value Pr > |t| 11 -1.09 0.2992 Mean of the differences P =0.3, suggest the mean of the difference is equal to 0 T statistics for testing if the mean of the difference is 0

  32. Two independent samples t-test • An independent samples t-test is used when you want to compare the means of a normally distributed interval dependent variable for two independent groups.  For example, using the hsb2 data file, say we wish to test whether the mean for write is the same for males and females.   proc ttestdata ="c:\hsb2"; classfemale; varwrite; run;

  33. Proc corr The CORR procedure is a statistical procedure for numeric random variables that computes correlation statistics (The default correlation analysis includes descriptive statistics, Pearson correlation statistics, and probabilities for each analysis variable). PROC CORRoptions; VAR variables; WITH variables; BY variables; Proc corr data=blood; var RBC WBC cholesterol; run;

  34. Proc Corr Output Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum RBC 908 7043 1003 6395020 4070 10550 WBC 916 5.48353 0.98412 5023 1.71000 8.75000 cholesterol 795 201.43522 49.88672 160141 17.00000 331.00000 N - This is the number of valid (i.e., non-missing) cases used in the correlation.  By default, proc corr uses pairwise deletion for missing observations, meaning that a pair of observations (one from each variable in the pair being correlated) is included if both values are non-missing.  If you use the nomiss option on the proc corr statement, proc corr uses listwise deletion and omits all observations with missing data on any of the named variables.

  35. Proc Corr Pearson Correlation Coefficients - measure the strength and direction of the linear relationship between the two variables.  The correlation coefficient can range from -1 to +1, with -1 indicating a perfect negative correlation, +1 indicating a perfect positive correlation, and 0 indicating no correlation at all.  Pearson Correlation Coefficients Prob > |r| under H0: Rho=0 Number of Observations RBC WBC cholesterol RBC 1.00000 -0.00203 0.06583 0.9534 0.0765 908 833 725 WBC -0.00203 1.00000 0.02496 0.9534 0.5014 833 916 728 cholesterol 0.06583 0.02496 1.00000 0.0765 0.5014 725 728 795 P-value Number of observations

  36. Proc Reg • The REG procedure is one of many regression procedures in the SAS System. PROC REG < options > ; MODEL dependents=<regressors> < / options > ; BY variables ; OUTPUT < OUT=SAS-data-set > keyword=names ;

  37. data blood; INFILE ‘F:\blood.txt'; INPUTsubjectID $ gender $ bloodtype $ age_group $ RBC WBC cholesterol; run; data blood1; set blood; if gender='Female' then sex=1; else sex=0; ifbloodtype='A' thentypeA=1; elsetypeA=0; if bloodtype='B' thentypeB=1; elsetypeB=0; if bloodtype='AB' then typeAB=1; elsetypeAB=0; ifage_group='Old' thenAge_old=1; elseAge_old=0; run; proc reg data =blood1; model cholesterol =sex typeAtypeBtypeABAge_old RBC WBC ; run;

  38. Proc reg output Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 41237 5891.02895 2.54 0.0140 Error 655 1521839 2323.41811 Corrected Total 662 1563076 DF - These are the degrees of freedom associated with the sources of variance.   (1) The total variance has N-1 degrees of freedom (663-1=662).   (2) The model degrees of freedom corresponds to the number of predictors minus 1 (P-1).  Including the intercept, there are 8 predictors, so the model has 8-1=7 degrees of freedom.  (3) The Residual degrees of freedom is the DF total minus the DF model, 662-7 is 655.

  39. Proc reg output Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 41237 5891.02895 2.54 0.0140 Error 655 1521839 2323.41811 Corrected Total 662 1563076 Sum of Squares - associated with the three sources of variance, total, model and residual. SSTotal     The total variability around the mean.  Sum(Y - Ybar)2.   SSResidual  The sum of squared errors in prediction.  Sum(Y - Ypredicted)2. SSModel     The improvement in prediction by using the predicted value of Y over just using the mean of Y. Hence, this would be the squared differences between the predicted value of Y and the mean of Y, Sum (Ypredicted - Ybar)2. Note that the SSTotal = SSModel + SSResidual.  SSModel / SSTotal is equal to the value of R-Square, the proportion of the variance explained by the independent variables

  40. Proc reg output Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 41237 5891.02895 2.54 0.0140 Error 655 1521839 2323.41811 Corrected Total 662 1563076 Mean Square - These are the Mean Squares, the Sum of Squares divided by their respective DF.  These are computed so you can compute the F ratio, dividing the Mean Square Model by the Mean Square Residual to test the significance of the predictors in the model

  41. Proc reg output Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 41237 5891.02895 2.54 0.0140 Error 655 1521839 2323.41811 Corrected Total 662 1563076 F Value and Pr > F - The F-value is the Mean Square Model divided by the Mean Square Residual.  F-value and P value are used to answer the question "Do the independent variables predict the dependent variable?".  The p-value is compared to your alpha level (typically 0.05) and, if smaller, you can conclude "Yes, the independent variables reliably predict the dependent variable".    Note that this is an overall significance test assessing whether the group of independent variables when used together reliably predict the dependent variable, and does not address the ability of any of the particular independent variables to predict the dependent variable. 

  42. Proc reg output Root MSE 48.20185 R-Square 0.0264 Dependent Mean 201.69683 Adj R-Sq 0.0160 CoeffVar 23.89817 Root MSE - Root MSE is the standard deviation of the error term, and is the square root of the Mean Square Residual (or Error).

  43. Proc reg output Root MSE 48.20185 R-Square 0.0264 Dependent Mean 201.69683 Adj R-Sq 0.0160 CoeffVar 23.89817 Dependent Mean - This is the mean of the dependent variable. CoeffVar - This is the coefficient of variation, which is a unit-less measure of variation in the data.  It is the root MSE divided by the mean of the dependent variable, multiplied by 100: (100*(48.2/201.69) =23.90).

  44. Proc reg output t Value and Pr > |t|- These columns provide the t-value and 2 tailed p-value used in testing the null hypothesis that the coefficient/parameter is 0. Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 187.91927 17.45409 10.77 <.0001 sex 1 1.48640 3.79640 0.39 0.6955 typeA 1 0.74839 4.01841 0.19 0.8523 typeB 1 10.14482 6.97339 1.45 0.1462 typeAB 1 -19.90314 10.45833 -1.90 0.0575 Age_old 1 -11.61798 3.85823 -3.01 0.0027 RBC 1 0.00264 0.00191 1.38 0.1676 WBC 1 0.20512 1.88816 0.11 0.9135

  45. ANOVA • A one-way analysis of variance (ANOVA) is used when you have a categorical independent variable (with two or more categories) and a normally distributed interval dependent variable and you wish to test for differences in the means of the dependent variable broken down by the levels of the independent variable.    .

  46. ANOVA • The following example studies the effect of bacteria on the nitrogen content of red clover plants. The treatment factor is bacteria strain, and it has six levels. Five of the six levels consist of five different Rhizobiumtrifolii bacteria cultures combined with a composite of five Rhizobiummeliloti strains. The sixth level is a composite of the five Rhizobiumtrifolii strains with the composite of the Rhizobiummeliloti. Red clover plants are inoculated with the treatments, and nitrogen content is later measured in milligrams. title1 'Nitrogen Content of Red Clover Plants'; dataClover; input Strain $ Nitrogen @@; datalines; 3DOK1 19.4 3DOK1 32.6 3DOK1 27.0 3DOK1 32.1 3DOK1 33.0 3DOK5 17.7 3DOK5 24.8 3DOK5 27.9 3DOK5 25.2 3DOK5 24.3 3DOK4 17.0 3DOK4 19.4 3DOK4 9.1 3DOK4 11.9 3DOK4 15.8 3DOK7 20.7 3DOK7 21.0 3DOK7 20.5 3DOK7 18.8 3DOK7 18.6 3DOK13 14.3 3DOK13 14.4 3DOK13 11.8 3DOK13 11.6 3DOK13 14.2 COMPOS 17.3 COMPOS 19.4 COMPOS 19.1 COMPOS 16.9 COMPOS 20.8 ; run; proc anovadata = Clover; class strain; model Nitrogen = Strain; run; proc freq data = Clover; tables Strain; run;

  47. ANOVA • The test for Strain suggests that there are differences among the bacterial strains, but it does not reveal any information about the nature of the differences. Mean comparison methods can be used to gather further information.

  48. HLTH 653 • This is a required class • It is part of the qualifier exams

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