Applications of Non-Parametric Test To The Analysis of Eye Disease Experimental Data
Introduction
The analyzed data was obtained from an experiment, 22 patients participated in the experiment, the treatment was applied to 11 patients with a specific eye infection in either eye (Right Eye:RE, Left Eye:LE, No Infection: OK) and 11 patients with no infection. The following responses - SEWR, NVA, CS and SA where collected before treatment and continuously for 8 weeks after the treatment was applied. The aim of this analysis was to determine if the treatment had a statistically significant effect on the measured scores.
After conducting exploratory data analysis, it was observed that the data is highly non-normal. The utilization of transformations with mixed model analysis did not produce well fitted models, also a multivariate non-parametric model was utilized but it did not produce efficient results. Therefore, the responses were broken down and tested separately using non-parametric tests.
First, a Friedman test was used to test each response and the responses with significant change were put through a post-hoc test using the Wilcox Rank Sum test, to investigate if the difference between pre and post test scores is significant.Finally a correlation analysis utilizing the Kendal Rank Correlation was performed to test the strength of association between the responses and Age of the patients.
Exploratory Data Analysis
Prior to performing the tests, the data were checked for completeness and explored to understand patterns in the data.
Data Cleaning
Read in the data.
week Case Age Sex GR AME
Min. :0 Min. : 1.0 Min. :18.00 : 0 : 0 : 0
1st Qu.:2 1st Qu.: 6.0 1st Qu.:19.00 F: 63 C:99 LE:45
Median :4 Median :11.5 Median :23.50 M:135 E:99 OK:99
Mean :4 Mean :11.5 Mean :24.95 RE:54
3rd Qu.:6 3rd Qu.:17.0 3rd Qu.:33.00
SEWRRE SEWRLE NVARE NVALE
Min. :-5.2500 Min. :-10.3700 Min. :0.2000 Min. :0.2000
1st Qu.:-1.5000 1st Qu.: -2.0000 1st Qu.:0.3000 1st Qu.:0.3000
Median : 0.2500 Median : 0.2500 Median :0.4000 Median :0.4000
Mean : 0.1693 Mean : -0.5176 Mean :0.3879 Mean :0.3697
3rd Qu.: 0.6200 3rd Qu.: 0.6200 3rd Qu.:0.4000 3rd Qu.:0.4000
CSRE CSLE SA
Min. :1.320 Min. :1.320 Min. : 1.6
1st Qu.:1.600 1st Qu.:1.600 1st Qu.: 50.0
Median :1.640 Median :1.680 Median : 60.0
Mean :1.623 Mean :1.628 Mean :136.9
3rd Qu.:1.680 3rd Qu.:1.680 3rd Qu.:140.0
[ reached getOption("max.print") -- omitted 1 row ]Profile Plots
A profile plot is a line plot used to visualize within subject factors and between subject factors in a repeated measures data. The lines represent the mean response of the subjects, the colors represent each infection group and the data was plotted for each variable over time (9 weeks).
SEWR
Change in SEWR for both the left and right eye in all groups was not noticable.
Right Eye
Left Eye
NVA
The scores for both the left and right eye did not change much, they ended up in a range between 0.3 to 0.6,
Right Eye
Left Eye
CS
There was an upward increase in CS scores - patients with infection in their right eye had an increase in CS scores in the right eye by the end of treatment and patients with infection in their left eye also had an increase in CS scores in their left eye.
Right Eye
Left Eye
SA
Patients with eye infection saw a decrease in SA scores, scores for patients with no infection remained constant.
SA
Univariate and Bivarate Graphs
Histograms of the responses were plotted with all the 3 (RE, LE, OK) groups together, and then seperately to visualize the distribution and symmetry of the data. From the plots it was clear that the responses are not normaly distributed, the CSRE and CSLE have very skewed left distributions and all the other responses either have right skewed distrutions and/or the plots are not continous, almost having the look of discrete/count data.
Histograms - All Groups Together
Histograms: Infected Right Eye
Histograms: Infected Left Eye
Histogram: No Infection
Pairplots
Below is a Pairplot to show the relationship between the continous variables. As can be seen there is no clear linear or non-linear relationship between the independent variable Age and the responses. A correlation analysis would determine if there is a statistical association between the responses and age, and the magnitude of the association if it does exist.
Non-Parametric Tests
Non-parametric tests are a class of statistical distribution free tests that are used in place of parametric tests when the data does not follow the requirements to use parametric test.
Mixed models are used to analyze repeated measures data. They include a fixed effects part of the model which includes factors being analyzed and of intrest to the researcher, and a random effects part which includes subject factors that are not of interest for analysis but need to be accounted for. In this case, the fixed effects are the infection groups, time taken (weeks), and the interaction between the time and groups. The random effects are each of the 22 patients.
The fixed effects section is the regular linear model and as such it has requirements for residual heterosckedacity and normality. The responses in this data are non-normal and they are not resolved with data transformations - a sample of the model for CS for the right eye is shown below with the residual plots. As such a Friedman test which is a non-parametric repeated measures test was used instead.
Mixed Model
Linear mixed-effects model fit by REML
Data: df
AIC BIC logLik
-951.6027 -925.5427 483.8013
Random effects:
Formula: ~1 | Case
(Intercept) Residual
StdDev: 0.02014465 0.01568935
Fixed effects: lgCSRE ~ AME + week + AME * week
Value Std.Error DF t-value p-value
(Intercept) 0.4843852 0.009987985 173 48.49679 0.0000
AMEOK 0.0254126 0.012045963 19 2.10963 0.0484
AMERE -0.1311445 0.013523795 19 -9.69731 0.0000
week 0.0035241 0.000905825 173 3.89044 0.0001
AMEOK:week -0.0029327 0.001092466 173 -2.68445 0.0080
AMERE:week 0.0122297 0.001226493 173 9.97131 0.0000
Correlation:
(Intr) AMEOK AMERE week AMEOK:
AMEOK -0.829
AMERE -0.739 0.612
week -0.363 0.301 0.268
AMEOK:week 0.301 -0.363 -0.222 -0.829
AMERE:week 0.268 -0.222 -0.363 -0.739 0.612
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-4.24030000 -0.45568589 0.02134148 0.58212612 2.63410188
Number of Observations: 198
Number of Groups: 22 
Friedman Test
The Friedman test is a non-parametric test, that tests for differences across multiple treatments by first ranking the blocks of data and analyzing them by columns. The data was divided into 3 groups corresponding to the eye infection groups (RE, LE, OK) and then the test was performed on each response.
The hypothesis test is as follows;
H0: The scores across the 9 weeks are equal
H1: The scores across the 9 weeks are different
All tests were evaluated at an alpha <= 0.05, anything greater was rejected as not being statistically significant.
SEWR/RE
RESULT:
RE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the right eye
LE: Do not reject null hypothesis - treatment scores across the 9 weeks are the same for patients with infection in the left eye
OK: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: SEWRRE and week and Case
Friedman chi-squared = 24, df = 8, p-value = 0.002292[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: SEWRRE and week and Case
Friedman chi-squared = 0, df = 8, p-value = 1[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: SEWRRE and week and Case
Friedman chi-squared = 16, df = 8, p-value = 0.04238SEWR/LE
RESULT:
RE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the right eye
LE: Do not reject null hypothesis - treatment scores across the 9 weeks are the same for patients with infection in the left eye
OK: Reject null hypothesis - treatment scores across the 9 weeks are not the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: SEWRLE and week and Case
Friedman chi-squared = 8, df = 8, p-value = 0.4335[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: SEWRLE and week and Case
Friedman chi-squared = 8, df = 8, p-value = 0.4335[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: SEWRRE and week and Case
Friedman chi-squared = 16, df = 8, p-value = 0.04238NVA/RE
RESULT:
RE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the right eye
LE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the left eye
OK: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: NVARE and week and Case
Friedman chi-squared = 24.076, df = 8, p-value = 0.002225[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: NVARE and week and Case
Friedman chi-squared = 16.333, df = 8, p-value = 0.03785[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: NVARE and week and Case
Friedman chi-squared = 17.73, df = 8, p-value = 0.02335NVA/LE
RESULT:
RE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the right eye
LE: Do not reject null hypothesis - treatment scores across the 9 weeks are the same for patients with infection in the left eye
OK: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: NVALE and week and Case
Friedman chi-squared = 8.8571, df = 8, p-value = 0.3545[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: NVALE and week and Case
Friedman chi-squared = 10.867, df = 8, p-value = 0.2093[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: NVALE and week and Case
Friedman chi-squared = 18.051, df = 8, p-value = 0.02085CS/LE
RESULT:
RE: Do not reject the null hypothesis - treatment scores across the 9 weeks are the same for patients with infection in the right eye
LE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the left eye
OK: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: CSLE and week and Case
Friedman chi-squared = 0, df = 8, p-value = 1[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: CSLE and week and Case
Friedman chi-squared = 35.191, df = 8, p-value = 2.468e-05[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: CSLE and week and Case
Friedman chi-squared = 18.404, df = 8, p-value = 0.01839CS/RE
RESULT:
RE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the right eye
LE: Reject null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the left eye
OK: Do not reject null hypothesis - treatment scores across the 9 weeks are the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: CSRE and week and Case
Friedman chi-squared = 45.162, df = 8, p-value = 3.429e-07[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: CSRE and week and Case
Friedman chi-squared = 13.684, df = 8, p-value = 0.09038[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: CSRE and week and Case
Friedman chi-squared = 11.077, df = 8, p-value = 0.1974SA
RESULT:
RE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the right eye
LE: Reject the null hypothesis - treatment scores across the 9 weeks are not the same for patients with infection in the left eye
OK: Do not reject null hypothesis - treatment scores across the 9 weeks are the same for patients with no infection
[1] "Friedman Test For Patients with Infection in the Right Eye"
Friedman rank sum test
data: SA and week and Case
Friedman chi-squared = 42.959, df = 8, p-value = 8.943e-07[1] "Friedman Test For Patients with Infection in the Left Eye"
Friedman rank sum test
data: SA and week and Case
Friedman chi-squared = 31.645, df = 8, p-value = 0.0001078[1] "Friedman Test For Patients with No Infection"
Friedman rank sum test
data: SA and week and Case
Friedman chi-squared = 3.375, df = 8, p-value = 0.9087Post-Hoc Test: Wilcox Test
The Friedman test is the non-parametric alternative to the repeated measures ANOVA. It tests for a difference between treatments but does not identify which pair has the difference, for this a post-hoc test needs to be performed. In this case we utilize the pairwise Wilcox signed test which is the non-parametric alternative to the paired t-test. It will be used to determine if the treatment was effective in making a change between week0 scores (pre-treatment) and and week8 scores ( post-treatment).
The profile plots give an idea of how the values changed overall for each response, these were used as a guide for creating the hypothesis test.
SEWR/RE
The changes from the profile plot where slight and so we test if there was any significant overall change between pre and post test scores.
H0: Pre_SEWRRE = Post_SEWRRE
H1: Pre_SEWRRE != Post_SEWRRE
RESULT:
RE:Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the right eye
LE:Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the left eye
OK:Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_SEWRRE and re$SEWRRE
V = 0, p-value = 0.1736
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_SEWRRE and le$SEWRRE
V = 6, p-value = 0.8501
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_SEWRRE and ok$SEWRRE
V = 3, p-value = 0.3711
alternative hypothesis: true location shift is not equal to 0SEWR/LE
The changes from the profile plot where slight and so we test if there was any significant overall change between pre and post test scores
H0: Pre_SEWRLE = Post_SEWRLE
H1: Pre_SEWRLE != Post_SEWRLE
RESULT:
RE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the right eye
LE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the left eye
OK: Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_SWERLE and re$SEWRLE
V = 1, p-value = 1
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_SWERLE and le$SEWRLE
V = 9, p-value = 0.2012
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_SWERLE and ok$SEWRLE
V = 3, p-value = 0.3711
alternative hypothesis: true location shift is not equal to 0NVA/RE
The changes from the profile plot was unclear
H0: Pre_NVARE = Post_NVARE
H1: Pre_NVARE != Post_NVARE
RESULT:
RE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the right eye
LE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the left eye
OK: Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_NVARE and re$NVARE
V = 0, p-value = 0.09467
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_NVARE and le$NVARE
V = 3, p-value = 0.3458
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_NVARE and ok$NVARE
V = 6, p-value = 0.1736
alternative hypothesis: true location shift is not equal to 0NVA/LE
The changes from the profile plot was unclear
H0: Pre_NVALE = Post_NVALE
H1: Pre_NVALE != Post_NVALE
RESULT:
RE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the right eye
LE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the left eye
OK: Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_NVALE and re$NVALE
V = 1, p-value = 1
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_NVALE and le$NVALE
V = 1.5, p-value = 1
alternative hypothesis: true location shift is not equal to 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_NVALE and ok$NVALE
V = 10, p-value = 0.08897
alternative hypothesis: true location shift is not equal to 0CS/RE
From the profile plot the experimental group had an increase in scores and the control group had no change
Experiment group
H0: Pre_CSRE <= Post_CSRE
H1: Pre_CSRE > Post_CSRE
Control group
H0: Pre_CSRE = Post_CSRE
H1: Pre_CSRE != Post_CSRE
RESULT:
RE: Reject null hypothesis - pre and post treatment scores are not the same for patients with infection in the right eye
LE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the left eye
OK: Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_CSRE and re$CSRE
V = 21, p-value = 0.01751
alternative hypothesis: true location shift is greater than 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_CSRE and le$CSRE
V = 1, p-value = 0.5
alternative hypothesis: true location shift is greater than 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_CSRE and ok$CSRE
V = 1, p-value = 1
alternative hypothesis: true location shift is not equal to 0CS/LE
From the profile plot the experimental group had an increase in scores
Experiment group
H0: Pre_CSLE <= Post_CSLE
H1: Pre_CSLE > Post_CSLE
Control group
H0: Pre_CSLE = Post_CSLE
H1: Pre_CSLE != Post_CSLE
RESULT:
RE: Do not reject null hypothesis - pre and post treatment scores are the same for patients with infection in the right eye
LE: Reject null hypothesis - pre and post treatment scores are not the same for patients with infection in the left eye
OK: Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_CSLE and re$CSLE
V = 0, p-value = 1
alternative hypothesis: true location shift is greater than 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_CSLE and le$CSLE
V = 15, p-value = 0.02895
alternative hypothesis: true location shift is greater than 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_CSLE and ok$CSLE
V = 1.5, p-value = 1
alternative hypothesis: true location shift is not equal to 0SA
From the profile plot the experimental group had a decrease in scores for the experimental group
Experiment Group
H0: Pre_SA >= Post_SA
H1: Pre_SA < Post_SA
Control Group
H0: Pre_SA = Post_SA
H1: Pre_SA != Post_SA
RESULT:
RE: Reject null hypothesis - pre and post treatment scores are not the same for patients with infection in the right eye
LE: Reject null hypothesis - pre and post treatment scores are not the same for patients with infection in the left eye
OK: Do not reject null hypothesis - pre and post treatment scores are the same for patients with no infection
[1] "Wilcox Test For Group with Infection in Right Eye"
Wilcoxon signed rank test with continuity correction
data: re$Post_SA and re$SA
V = 0, p-value = 0.01751
alternative hypothesis: true location shift is less than 0[1] "Wilcox Test For Group with Infection in Left Eye"
Wilcoxon signed rank test with continuity correction
data: le$Post_SA and le$SA
V = 0, p-value = 0.02724
alternative hypothesis: true location shift is less than 0[1] "Wilcox Test For Group No Infection"
Wilcoxon signed rank test with continuity correction
data: ok$Post_SA and ok$SA
V = 1.5, p-value = 1
alternative hypothesis: true location shift is not equal to 0Correlation Analysis
To determine if there is an association between Age and the recorded responses, a Kendall rank correlation was used. The Kendall Rank correlation is a non-parametric test that measures the strength of dependence between two continous variables. It was used in this analysis because the test has no distribution or independence requirements.
H0: tau = 0
H1: tau != 0
RIGHT EYE INFECTED
SEWR/RE: Fail to reject the null hypothesis - Age and SEWR in the right eye are independent for patients with infection in the right eye
Kendall's rank correlation tau
data: re$Age and re$SEWRRE
z = 0.39632, p-value = 0.6919
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.1482499 SWER/LE: Reject the null hypothesis - Age and SEWR in the right eye are dependent for patients with infection in the right eye.
There is a weak negative association
Kendall's rank correlation tau
data: re$Age and re$SEWRLE
z = -0.58461, p-value = 0.5588
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.2148345 NVA/RE: Fail to reject the null hypothesis - Age and NVA in the right eye are independent for patients with infection in the right eye
Kendall's rank correlation tau
data: re$Age and re$NVARE
z = 0, p-value = 1
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0 NVA/LE: Reject the null hypothesis - Age and NVA in the left eye are dependent for patients with infection in the right eye.
There is a medium negative association
Kendall's rank correlation tau
data: re$Age and re$NVALE
z = -0.95346, p-value = 0.3404
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.3922323 CS/RE: Reject the null hypothesis - Age and CS in the right eye are dependent for patients with infection in the right eye.
There is a medium negative association
Kendall's rank correlation tau
data: re$Age and re$CSRE
z = -1.3641, p-value = 0.1725
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.5012804 CS/LE: Fail to reject the null hypothesis - Age and CS in the left eye are independent for patients with infection in the right eye
Kendall's rank correlation tau
data: re$Age and re$CSLE
z = 0, p-value = 1
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0 SA: Fail to reject the null hypothesis - Age and SA eye are independent for patients with infection in the right eye
Kendall's rank correlation tau
data: re$Age and re$SA
z = -0.69812, p-value = 0.4851
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.2773501 LEFT EYE INFECTED
SEWR/RE: Reject the null hypothesis - Age and SEWR in the right eye are dependent for patients with infection in the left eye There is a strong positive association
Kendall's rank correlation tau
data: le$Age and le$SEWRRE
z = 1.2632, p-value = 0.2065
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.5270463 SWER/LE: Reject the null hypothesis - Age and SEWR in the right eye are dependent for patients with infection in the left eye.
There is a strong positive association
Kendall's rank correlation tau
data: le$Age and le$SEWRLE
T = 8, p-value = 0.2333
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.6 NVA/RE: Reject the null hypothesis - Age and NVA in the right eye are dependent for patients with infection in the left eye There is a strong negative association
Kendall's rank correlation tau
data: le$Age and le$NVARE
z = -0.70711, p-value = 0.4795
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.3162278 NVA/LE: Reject the null hypothesis - Age and NVA in the left eye are dependent for patients with infection in the right eye.
There is a medium negative association
Kendall's rank correlation tau
data: re$Age and re$NVALE
z = -0.95346, p-value = 0.3404
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.3922323 CS/RE: Reject the null hypothesis - Age and CS in the right eye are dependent for patients with infection in the left eye.
There is a medium positive association
Kendall's rank correlation tau
data: le$Age and le$CSRE
z = 1.9415, p-value = 0.0522
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.83666 CS/LE: Fail to reject the null hypothesis - Age and CS in the left eye are independent for patients with infection in the leftt eye
Kendall's rank correlation tau
data: le$Age and le$CSLE
z = 0.75794, p-value = 0.4485
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.3162278 SA: Fail to reject the null hypothesis - Age and SA eye are independent for patients with infection in the right eye
Kendall's rank correlation tau
data: re$Age and re$SA
z = -0.69812, p-value = 0.4851
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.2773501 NO INFECTION
SEWR/RE: Fail to reject the null hypothesis - Age and SEWR in the right eye are independent for patients with no infection
Kendall's rank correlation tau
data: ok$Age and ok$SEWRRE
z = 0.55614, p-value = 0.5781
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.1347151 SWER/LE: Reject the null hypothesis - Age and SEWR in the left eye are dependent for patients with no infection
There is a medium positive association
Kendall's rank correlation tau
data: ok$Age and ok$SEWRLE
z = 1.8161, p-value = 0.06935
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.4340395 NVA/RE: Fail to reject the null hypothesis - Age and NVA in the right eye are independent for patients with no infection
There is a medium positive association
Kendall's rank correlation tau
data: ok$Age and ok$NVARE
z = 1.3901, p-value = 0.1645
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.3666178 NVA/LE: Reject the null hypothesis - Age and NVA in the left eye are dependent for patients with no infection.
There is a medium positive association
Kendall's rank correlation tau
data: ok$Age and ok$NVALE
z = 0.60549, p-value = 0.5449
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.1601282 CS/RE: Reject the null hypothesis - Age and CS in the right eye are dependent for patients with no infection
There is a weak negative association
Kendall's rank correlation tau
data: ok$Age and ok$CSRE
z = -0.26079, p-value = 0.7943
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
-0.06711561 CS/LE: Fail to reject the null hypothesis - Age and CS in the left eye are independent for patients with no infection
Kendall's rank correlation tau
data: ok$Age and ok$CSLE
z = 0.40918, p-value = 0.6824
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.1111111 SA: Fail to reject the null hypothesis - Age and SA eye are independent for patients with infection in the right eye
Kendall's rank correlation tau
data: ok$Age and ok$SA
z = 0.60549, p-value = 0.5449
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.1601282 Conclusion
A mixed model was initially utilized but since the data was highly non-normal, data transformations did not produce well fitted models. Instead, the Friedman test was used to determine if there was a difference among the three infection groups. Then a Post-hoc test was performed using the Wilcox test to ascertain if there was a difference between pre and post treatment scores.
The result of the tests was that there was an increase between pre and post treatment scores for: 1. CS in the right eye for patients with infection in either eye. 2. CS in the left eye for patients with infection in the left eye. 3. SA for patients with infection in either eye. The treatment can therefore be said to be effective for the responses listed above.
The placebo effect would provide an explanation for changes in scores for patients without an eye infection, especially since all the Wilcox test for this group of patients did not reject the null that there was no statistical change in this group.
To check if there is an association between age and the responses, and the magnitude of the association, a Kendall Rank correlation test was utilized across the 3 infection groups. The results across groups was not consistent and so more data will need to be collected and analyzed to make a consistent conclusion.