Download Statistical Analysis of Data: Detection Thresholds for 0.025 Contrast

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/m2. The other line has a
background luminance of 4 cd/m2 and a target luminance of 4.1 cd/m2. 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/m2 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
taken for each subject. t-tests allow for inference about the mean without
2
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 Lum12   BG Lum 4
H a :  BG Lum12   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
Detection Thresholds for 0.025 Contrast Levels
16.00
14.00
Distance (ft)
12.00
10.00
BG Lum 12 cd/m^2
8.00
BG Lum 4 cd/m^2
6.00
4.00
2.00
0.00
1
2
3
4
5
6
7
8
Subject
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;
proc print data = contrast;
run;
4
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.
t
d  0
(Equation 1.2)
sd / n
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
Rectangular Detection Thresholds
20
18
Distance (ft)
16
14
12
BG Lum 12
10
BG Lum 4
8
6
4
2
0
0
0.01
0.02
0.03
0.04
Contrast (cd/m^2)
Figure 1.2: Detection Thresholds for Increasing Contrast Levels
The data analyzed in this paper was 0.025 and the mean is represented as the
two points where x = 0.025. By just looking at this graph, it appears that a higher
background luminance increases the distance from which the stimuli can be
detected. More generally, the brighter the overall picture, the further away a line
can be seen. However, from the statistical analysis, the null hypothesis cannot
be rejected for a 95% confidence level. This is most likely due to the very small
sample size, which does contain outliers. This graph is a testament to the need
for both an appropriate sample size and statistical analysis.
6