Frances Wu 22S 30 Spring 2004 Statistical Analysis of Data Detection Thresholds for 0 025 Contrast Level When collecting data statistical analysis is necessary to determine the significance of the data This is used to establish whether or not the data holds for a certain hypothesis For example data is being collected in an observational study for Ergonomics 56 147 and must undergo statistical analysis This study Detection Thresholds for Rectangular Targets with Different Aspect Ratios and Orientations is trying to determine if a difference exists for the detection of stimuli with different contrasts lengths widths and orientations The research question proposed in this paper is Given two lines with the same contrast level is there an effect on the detection threshold if the overall luminance is higher or lower Data collection for the study is an ongoing process and has only resulted in eight sets of data at this point The testing procedure calls for an individual to walk towards a computer screen that is presenting a line at a very low contrast At the point where the subject can identify the orientation horizontal vertical or diagonal the distance from the screen is recorded A total of 81 runs are given from a combination of three widths three angle orientations and nine combinations of background and target luminance levels The data has been dramatically cut back for use in this statistical analysis to only two combinations of variables Two lines were chosen from the test to be used as the two treatments and the distances from which each subject detected the line as the 1 response variable The lines are both horizontal and have a contrast level of 0 025 One set of data is for the line with a background luminance BG Lum of 12 cd m2 and a target luminance TG Lum of 12 3 cd m 2 The other line has a background luminance of 4 cd m2 and a target luminance of 4 1 cd m 2 The contrast is determined by the equation Contrast BG Lum TG Lum BG Lum Equation 1 1 By using Equation 1 1 the contrast for both lines is determined to be 0 025 The eight sets of data collected are presented in Table 1 1 TABLE 1 1 Collected Data for 0 025 Contrast Levels BG Lum 12 ft BG Lum 4 ft Subject 1 13 50 4 00 Subject 2 3 50 6 75 Subject 3 7 50 1 00 Subject 4 10 00 1 75 Subject 5 12 75 9 25 Subject 6 1 25 5 50 Subject 7 10 00 2 75 Subject 8 4 00 1 75 This is the raw data that was used for further statistical analysis For simplicity the distance observations will be further referred to as BG Lum 12 and BG Lum 4 for background luminance 12 cd m 2 and background luminance 4 cd m2 respectively From a comparison of a variety of statistical tests such as z tests t tests and X2 tests a paired t test was chosen to be the best to analyze the data The data is self paired because the measurements for BG Lum 12 and BG Lum 4 were 2 taken for each subject t tests allow for inference about the mean without knowing the population standard of deviation The paired t test further allows for a comparison of the difference in the distances recorded for each subject This is beneficial because eyesight abilities vary greatly for each individual and the detection at an overall brighter luminance can be compared with a lower luminance target The difference was calculated by subtracting the distance for BG Lum 4 from the distance for BG Lum 12 The mean of these distances were then used for calculating the t statistic A two sided test is completed because it is not initially evident whether detection is easier at a higher or lower luminance level The null hypothesis used was that there exists no difference in the detection thresholds for the BG Lum 12 and BG Lum 4 H 0 BG Lum 12 BG Lum 4 H a BG Lum 12 BG Lum 4 The robustness for t tests does not hold for data sets that contain outliers The effects of outliers are enhanced even more when the sample size is small By the central limit theorem distribution approaches Normal for sample means as the sample size increases In this case the sample size is only 8 More data is undeniably needed for this analysis but is not available at this time due to the concurrent progress for both projects A graphical representation of the data collected is shown on the following page in Figure 1 1 As a rule of thumb t tests should only be conducted for sample sizes less than 15 if the data is near Normal contains no outliers and is not skewed For sample sizes greater than 15 the test still cannot be used with outliers Because more data is not available at this time t tests were still conducted as the best test for this situation 3 Figure 1 1 Detection Thresholds for 0 025 Contrast Levels SAS was used for the statistical analysis of the data testing for a 0 05 significance level The following program shows the input used to run the paired t tests options linesize 79 data contrast input twelve four 8 diff twelve four datalines 13 50 4 00 3 50 6 75 7 50 1 00 10 00 1 75 12 75 9 25 1 25 5 50 10 00 2 75 4 00 1 75 run 4 proc print data contrast run proc means data contrast n mean stdev stderr clm alpha 0 05 var diff run proc univariate data contrast var diff run Appendix A shows the complete output generated from this program The tstatistic can be calculated from the data produced by the proc means function using Equation 1 2 d 0 t sd n Equation 1 2 The mean calculated by SAS is 3 71875 with a standard deviation of 5 1883067 The t statistic then calculates to be 2 0273 The degrees of freedom for this set of data is n 1 7 From Table C the 95 confidence level for 7 degrees of freedom has t 2 365 Because the 2 0273 2 365 the data is determined to not be reliable enough for the 95 confidence level Therefore the null hypothesis cannot be rejected The information generated from proc univariate provides the same information just calculated The t statistic is given as 2 027292 and the p value is already calculated as 0 0822 Because 0 0822 0 05 the significance level being tested the hypothesis as already determined cannot be rejected A simple graph generated in Excel appears to relay convincing data The top line portrays the mean distances taken for a series of increasing contrasts with a background luminance of 12 and the bottom line has the same levels of increasing contrasts with a background luminance of 4 5 Figure 1 2 Detection Thresholds for Increasing Contrast Levels The data analyzed in this paper was 0 025 …