The SAS System 08:43 Thursday, January 23, 2014 1 The FREQ Procedure Table of treatment by response treatment response Frequency‚ Expected ‚ Percent ‚ Row Pct ‚ Col Pct ‚cold ‚nocold ‚ Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ ascorbic ‚ 17 ‚ 122 ‚ 139 ‚ 23.914 ‚ 115.09 ‚ ‚ 6.09 ‚ 43.73 ‚ 49.82 ‚ 12.23 ‚ 87.77 ‚ ‚ 35.42 ‚ 52.81 ‚ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ placebo ‚ 31 ‚ 109 ‚ 140 ‚ 24.086 ‚ 115.91 ‚ ‚ 11.11 ‚ 39.07 ‚ 50.18 ‚ 22.14 ‚ 77.86 ‚ ‚ 64.58 ‚ 47.19 ‚ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total 48 231 279 17.20 82.80 100.00 Statistics for Table of treatment by response Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1 4.8114 0.0283 Likelihood Ratio Chi-Square 1 4.8717 0.0273 Continuity Adj. Chi-Square 1 4.1407 0.0419 Mantel-Haenszel Chi-Square 1 4.7942 0.0286 Phi Coefficient -0.1313 Contingency Coefficient 0.1302 Cramer's V -0.1313 Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 17 Left-sided Pr <= F 0.0205 Right-sided Pr >= F 0.9910 Table Probability (P) 0.0115 Two-sided Pr <= P 0.0385 The SAS System 08:43 Thursday, January 23, 2014 2 The FREQ Procedure Statistics for Table of treatment by response Column 1 Risk Estimates (Asymptotic) 95% (Exact) 95% Risk ASE Confidence Limits Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Row 1 0.1223 0.0278 0.0678 0.1768 0.0729 0.1886 Row 2 0.2214 0.0351 0.1527 0.2902 0.1557 0.2993 Total 0.1720 0.0226 0.1278 0.2163 0.1297 0.2215 Difference -0.0991 0.0448 -0.1869 -0.0114 Difference is (Row 1 - Row 2) Column 2 Risk Estimates (Asymptotic) 95% (Exact) 95% Risk ASE Confidence Limits Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Row 1 0.8777 0.0278 0.8232 0.9322 0.8114 0.9271 Row 2 0.7786 0.0351 0.7098 0.8473 0.7007 0.8443 Total 0.8280 0.0226 0.7837 0.8722 0.7785 0.8703 Difference 0.0991 0.0448 0.0114 0.1869 Difference is (Row 1 - Row 2) The SAS System 08:43 Thursday, January 23, 2014 3 The FREQ Procedure Statistics for Table of treatment by response Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control (Odds Ratio) 0.4900 0.2569 0.9343 Cohort (Col1 Risk) 0.5523 0.3209 0.9506 Cohort (Col2 Risk) 1.1273 1.0120 1.2558 Odds Ratio (Case-Control Study) ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Odds Ratio 0.4900 Asymptotic Conf Limits 95% Lower Conf Limit 0.2569 95% Upper Conf Limit 0.9343 Exact Conf Limits 95% Lower Conf Limit 0.2407 95% Upper Conf Limit 0.9740 Sample Size = 279 The SAS System 08:43 Thursday, January 23, 2014 4 The FREQ Procedure Table of treatment by response treatment response Frequency‚ Percent ‚ Row Pct ‚ Col Pct ‚cold ‚nocold ‚ Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ absorbic ‚ 170 ‚ 1220 ‚ 1390 ‚ 6.09 ‚ 43.73 ‚ 49.82 ‚ 12.23 ‚ 87.77 ‚ ‚ 35.42 ‚ 52.81 ‚ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ placebo ‚ 310 ‚ 1090 ‚ 1400 ‚ 11.11 ‚ 39.07 ‚ 50.18 ‚ 22.14 ‚ 77.86 ‚ ‚ 64.58 ‚ 47.19 ‚ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total 480 2310 2790 17.20 82.80 100.00 Statistics for Table of treatment by response Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1 48.1141 <.0001 Likelihood Ratio Chi-Square 1 48.7170 <.0001 Continuity Adj. Chi-Square 1 47.4207 <.0001 Mantel-Haenszel Chi-Square 1 48.0969 <.0001 Phi Coefficient -0.1313 Contingency Coefficient 0.1302 Cramer's V -0.1313 Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 170 Left-sided Pr <= F 2.138E-12 Right-sided Pr >= F 1.0000 Table Probability (P) 1.105E-12 Two-sided Pr <= P 3.441E-12 Sample Size = 2790 The SAS System 08:43 Thursday, January 23, 2014 5 The FREQ Procedure Table of treatment by response treatment response Frequency‚ Percent ‚ Row Pct ‚ Col Pct ‚cold ‚nocold ‚ Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ absorbic ‚ 2 ‚ 12 ‚ 14 ‚ 7.14 ‚ 42.86 ‚ 50.00 ‚ 14.29 ‚ 85.71 ‚ ‚ 40.00 ‚ 52.17 ‚ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ placebo ‚ 3 ‚ 11 ‚ 14 ‚ 10.71 ‚ 39.29 ‚ 50.00 ‚ 21.43 ‚ 78.57 ‚ ‚ 60.00 ‚ 47.83 ‚ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total 5 23 28 17.86 82.14 100.00 Statistics for Table of treatment by response Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1 0.2435 0.6217 Likelihood Ratio Chi-Square 1 0.2448 0.6207 Continuity Adj. Chi-Square 1 0.0000 1.0000 Mantel-Haenszel Chi-Square 1 0.2348 0.6280 Phi Coefficient -0.0933 Contingency Coefficient 0.0928 Cramer's V -0.0933 WARNING: 50% of the cells have expected counts less than 5. Chi-Square may not be a valid test. Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 2 Left-sided Pr <= F 0.5000 Right-sided Pr >= F 0.8370 Table Probability (P) 0.3370 Two-sided Pr <= P 1.0000 The SAS System 08:43 Thursday, January 23, 2014 6 The FREQ Procedure Statistics for Table of treatment by response Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control (Odds Ratio) 0.6111 0.0854 4.3706 Cohort (Col1 Risk) 0.6667 0.1308 3.3979 Cohort (Col2 Risk) 1.0909 0.7709 1.5438 Odds Ratio (Case-Control Study) ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Odds Ratio 0.6111 Asymptotic Conf Limits 95% Lower Conf Limit 0.0854 95% Upper Conf Limit 4.3706 Exact Conf Limits 95% Lower Conf Limit 0.0441 95% Upper Conf Limit 6.5424 Sample Size = 28