Client: STAT 581 Case Study 2 Client
Consultant: Divya Kumar
Date: 08/04/14
Research field: Kinesiology
Project Title: Case Study 2 Kinesiology Experiment
Project description:
Eight human subjects were selected. The study was not randomized, and the subjects were not blinded. We assume that the subjects were right-handed male adults, as in the research paper provided to us for reference to a similar study.1 Your population of interest is adult humans.
A robot was used in the experiment, with which the human subjects interacted. The experiment took place on campus, at a motor control lab. The robot was programmed. Certain intentional movements of the human subjects were measured upon the subjects’ interaction with the robots. The subjects also performed unintentional movements when given a stimulus from the robot. There were six intentional movements. All subjects went through the six movements in the same order. The six movements were a combination of three directions of movement2, and intentional or unintentional movement. There are two different types of responses measured—one related to position, and the other related to orientation.
Note: I will set an alpha level of 0.05.
Research Question:
1 Unpredictable elbow joint perturbation during reaching results in multijoint motor equivalenceD. J. S. Mattos, M. L. Latash, E. Park, J. Kuhl, J. P. Scholz Journal of Neurophysiology Sep 2011,106(3)1424-1436; DOI:10.1152/jn.00163.20112 The three directions of movements are x+, y+, and y-, along a Cartesian plane.
1
1. How do movement type (intentional and unintentional) and movement direction affect the change in position, and also the change in orientation?
2. What does interaction mean in this experiment?
Statistical Questions:
1. Are the data normal? How would we test to check for normality? If the data are not normal, what implications does this have for this experiment?
Variables of Interest:
Explanatory Variables: x+, y+, y-, int3, uni4
Response Variable: delta_pos5, delta_ori6
Procedure:
We will use SPSS 21 for all of our analyses. But we will also do some data manipulation in Excel. Our procedure is:
1) We will test the stated null hypothesis using a two-way repeated measures ANOVA. With our two-way repeated measures ANOVA, we will test the hypotheses for both main and interaction effects. This will answer Research Questions 1 and 2.
Assumptions and Hypotheses:
Before we begin our first procedure, mentioned above, we need to check our assumptions for the two-way repeated measures ANOVA. The procedure being employed in the Repeated Measures procedure in SPSS is a multivariate test. This is one of three possible tests, the other two being a traditional univariate test, and the alternative univariate test with corrected degrees of freedom The procedure used by SPSS, namely the multivariate procedure, does not assume sphericity. Shpericity is an assumption similar to the homogeneity of variances in the between-subjects ANOVA, the “regular” ANOVA. Therefore, the sphericity assumption is the assumption of homogeneity of variances between all combinations of factor levels. Most applied statisticians choose the multivariate test over the others.
In SPSS using this procedure, difference scores are calculated between the means for each of the two types of movements (unintentional and intentional) over the three directions (y+, y-, x+). Two Excel tables are presented on the following page (page 3) with the data set up and the difference scores.For delta POS:
3 Intentional movement4 Unintentional movement5 Position6 Orientation
2
Subject1 0.63 0.32 0.15 0.58 0.85 0.72 0.29 0.29 0.47 0.92 0.39 0.243 0.54 0.44 0.3 1.19 0.72 0.764 0.51 0.56 0.26 0.91 0.81 0.485 0.35 0.26 0.02 0.33 0.23 0.296 0.35 0.62 0.33 0.94 0.84 0.847 0.66 0.16 0.23 1.05 1.01 0.898 0.62 0.6 0.4 0.52 0.61 0.67
POS_Uni_Ypos
POS_Uni_Yneg
POS_Uni_Xpos
POS_Int_Ypos
POS_Int_Yneg
POS_Int_Xpos
For delta POS:
Subject Means Int Difference1 0.366667 0.71 -0.343333332 0.35 0.516667 -0.166666673 0.426667 0.89 -0.463333334 0.443333 0.733333 -0.295 0.21 0.283333 -0.073333336 0.433333 0.873333 -0.447 0.35 0.983333 -0.633333338 0.54 0.6 -0.06
Means Uni
For delta ORI:
3
Subject1 0.649016 0.87428 0.894777 0.461444 0.165644 0.5224942 0.495006 0.573388 0.715803 0.3235 0.594481 0.4858363 0.266319 0.170448 0.644104 0.220635 0.425488 0.3361064 0.170573 0.336658 0.774616 0.477297 0.481047 0.4387395 0.517687 0.74292 0.764079 0.766485 0.501401 0.6189526 0.41004 0.465244 0.89587 0.528088 0.626879 0.2274037 0.942255 0.372623 0.799271 0.662423 1.066208 0.7978618 0.121386 0.458318 0.3915 0.625693 0.435636 0.481961
ORI_Uni_Ypos
ORI_Uni_Yneg
ORI_Uni_Xpos
ORI_Int_Ypos
ORI_Int_Yneg
ORI_Int_Xpos
For delta ORI:
Subject Means Int Difference1 0.806024 0.383194 0.42283032 0.594732 0.467939 0.1267933 0.36029 0.32741 0.03288054 0.427282 0.465695 -0.0384125 0.674895 0.628946 0.04594926 0.590385 0.46079 0.1295957 0.704716 0.842164 -0.1374488 0.323735 0.51443 -0.190695
Means Uni
S
Hypotheses for delta POS and delta ORI:
Our first hypothesis is that the population mean of the difference scores between intentional and unintentional movements across the three directions is zero.
Our second assumption is of the interaction effect. Our assumption is that the difference scores over the directions between the two types of movements is zero. Thus, we get three difference variables that we test on.
Back to our assumptions: there are two when using this multivariate method. The first is that the difference scores are multivariately normally distributed in the population. If it is not distrubted as such, and if the sample size is small7, our results may be invalid and/or the power of the test may suffer. The second assumption is that the sample being used is random, from the population, and the differences scores of any subject are independent from the difference scores of any other subject. It is very important to note here that if this assumption is violated, our results will not be useful. As we understand, the subjects in this experiment were not randomly chosen, nor were they randomly assigned. Unfortunately, this is a big shortcoming in making inferences to the
7 We understand that 8 subjects seems to be the norm for such studies. So, perhaps this will not be an issue.
4
population of adult males, which you wish to do. The results of this experiment, and this analysis, can only extend to the 8 subjects you tested under the same conditions they were tested.
Exploratory Data Analysis:
First we check, separately, to see if our dependent variables, delta POS and delta ORI, are normally distributed. This can be achieved by an examination of the QQ-plot.
For delta POS:
As the points fall mostly on the straight line, we can assume normality of delta POS.
For delta ORI:
5
Again, the points fall along the line of normality. So the normality assumption is not violated for delta ORI.
This gives us a preliminary idea of our assumptions. We will further explore normality below, when we examine several exploratory data analyses. These were conducted by selecting Analyze>Descriptive Statistics>Explore.
For delta POS:
6
Case Processing SummaryDirection of Movement
CasesValid Missing Total
N Percent N Percent N Percent
Delta POSX+ 16 100.0% 0 0.0% 16 100.0%Y- 16 100.0% 0 0.0% 16 100.0%Y+ 16 100.0% 0 0.0% 16 100.0%
This is a quality control check on our data. The N = 16 tells us that we entered our data correctly. (8 subjects * 2 directions in each of unintentional and intentional movements)
Next are several descriptive statistics, over each direction of movment:Descriptives
Direction of Movement Statistic Std. Error
Delta POS
X+
Mean .4392 .06524
95% Confidence Interval for Mean
Lower Bound
.3002
Upper Bound
.5783
5% Trimmed Mean .4378Median .3613Variance .068Std. Deviation .26094Minimum .02Maximum .89Range .87Interquartile Range .45Skewness .384 .564Kurtosis -.945 1.091
Y- Mean .5432 .06391
95% Confidence Interval for Mean
Lower Bound
.4069
Upper Bound
.6794
5% Trimmed Mean .5387
7
Median .5795Variance .065Std. Deviation .25565Minimum .16Maximum 1.01Range .85Interquartile Range .49Skewness .167 .564Kurtosis -1.095 1.091
Y+
Mean .6499 .06888
95% Confidence Interval for Mean
Lower Bound
.5031
Upper Bound
.7967
5% Trimmed Mean .6398Median .6029Variance .076Std. Deviation .27554Minimum .29Maximum 1.19Range .90Interquartile Range .53Skewness .517 .564Kurtosis -.771 1.091
Here we can see that the skewness and kurtosis statistics are close to 0, indicating a Gaussian distribution.
Below, is a test for normality:
8
Tests of NormalityDirection of Movement
Kolmogorov-Smirnova Shapiro-WilkStatistic df Sig. Statistic df Sig.
Delta POSX+ .168 16 .200* .939 16 .335Y- .125 16 .200* .955 16 .568Y+ .167 16 .200* .927 16 .221
*. This is a lower bound of the true significance.a. Lilliefors Significance Correction
All p-values indicate that the assumption of normality has not been violated.
Case Processing SummaryType of Movement
CasesValid Missing Total
N Percent N Percent N Percent
Delta POSINT 24 100.0% 0 0.0% 24 100.0%UNI 24 100.0% 0 0.0% 24 100.0%
Similar to the quality control check for direction of movement, this is one for the type of movement (int = intentional and uni = unintentional). Our N = 24 is a good sign, since 8 subjects * 6 measurements / 2 = 24.
Again, descriptive statistics, this time for type of movement:
9
DescriptivesType of Movement Statistic Std.
Error
Delta POS
INT
Mean .6978 .05476
95% Confidence Interval for Mean
Lower Bound
.5845
Upper Bound
.8111
5% Trimmed Mean .6976Median .7417Variance .072Std. Deviation .26829Minimum .23Maximum 1.19Range .95Interquartile Range .41Skewness -.299 .472Kurtosis -.793 .918
UNI
Mean .3904 .03576
95% Confidence Interval for Mean
Lower Bound
.3164
Upper Bound
.4643
5% Trimmed Mean .3950Median .3516Variance .031Std. Deviation .17519Minimum .02Maximum .66Range .65Interquartile Range .28Skewness -.092 .472Kurtosis -.726 .918
Again, our skewness and kurtosis statistics look good.
10
And finally, for delta POS, our tests for normality indicate that normality has not been violated:
Tests of NormalityType of Movement
Kolmogorov-Smirnova Shapiro-WilkStatistic df Sig. Statistic df Sig.
Delta POSINT .116 24 .200* .958 24 .397UNI .124 24 .200* .955 24 .354
*. This is a lower bound of the true significance.a. Lilliefors Significance Correction
For delta ORI:
Case Processing SummaryDirection of Movement
CasesValid Missing Total
N Percent N Percent N Percent
Delta ORI
X+ 16 100.0% 0 0.0% 16 100.0%Y- 16 100.0% 0 0.0% 16 100.0%Y+ 16 100.0% 0 0.0% 16 100.0%
Our quality control check for the direction of movement leads to a good conclusion, with N = 16.
Now, for our descriptive statistics for direction of movement:
DescriptivesDirection of Movement Statistic Std.
ErrorDelta ORI
X+ Mean .6118 .05144
95% Confidence Interval for Mean
Lower Bound
.5022
Upper Bound
.7215
5% Trimmed Mean .6174
11
Median .6315Variance .042Std. Deviation .20577Minimum .23Maximum .90Range .67Interquartile Range .34Skewness -.271 .564Kurtosis -1.031 1.091
Y-
Mean .5182 .05849
95% Confidence Interval for Mean
Lower Bound
.3935
Upper Bound
.6428
5% Trimmed Mean .5073Median .4731Variance .055Std. Deviation .23397Minimum .17Maximum 1.07Range .90Interquartile Range .23Skewness .751 .564Kurtosis .971 1.091
Y+ Mean .4774 .05589
95% Confidence Interval for Mean
Lower Bound
.3582
Upper Bound
.5965
5% Trimmed Mean .4713Median .4862Variance .050Std. Deviation .22356Minimum .12Maximum .94Range .82
12
Interquartile Range .36Skewness .240 .564Kurtosis -.197 1.091
Again, the skewness and kurtosis values are close to 0, indicating a Gaussian distribution.
Our tests of normality tell us that our assumption of the normal distribution is not violated:
Tests of NormalityDirection of Movement
Kolmogorov-Smirnova Shapiro-WilkStatistic df Sig. Statistic df Sig.
Delta ORI
X+ .145 16 .200* .950 16 .482Y- .154 16 .200* .943 16 .385Y+ .098 16 .200* .977 16 .939
*. This is a lower bound of the true significance.a. Lilliefors Significance Correction
Now, we do our quality control check for type of movement:
Case Processing SummaryType of Movement
CasesValid Missing Total
N Percent N Percent N PercentDelta ORI
INT 24 100.0% 0 0.0% 24 100.0%UNI 24 100.0% 0 0.0% 24 100.0%
Our N = 24 is good sign that we entered our values in correctly.
Now, for our descriptive statistics:
Descriptives
13
Type of Movement Statistic Std. Error
Delta ORI
INT
Mean .5113 .04034
95% Confidence Interval for Mean
Lower Bound
.4279
Upper Bound
.5948
5% Trimmed Mean .5017Median .4839Variance .039Std. Deviation .19764Minimum .17Maximum 1.07Range .90Interquartile Range .20Skewness .718 .472Kurtosis 1.637 .918
UNI
Mean .5603 .05091
95% Confidence Interval for Mean
Lower Bound
.4549
Upper Bound
.6656
5% Trimmed Mean .5634Median .5455Variance .062Std. Deviation .24942Minimum .12Maximum .94Range .82Interquartile Range .39Skewness -.157 .472Kurtosis -1.084 .918
Again, our skewness and kurtosis statistics are close to 0, indicating a Guassian distribution.
14
And finally, our tests for normality tell us that this assumption is not violated:
Tests of NormalityType of Movement
Kolmogorov-Smirnova Shapiro-WilkStatistic df Sig. Statistic df Sig.
Delta ORI
INT .133 24 .200* .949 24 .253UNI .109 24 .200* .954 24 .327
*. This is a lower bound of the true significance.a. Lilliefors Significance Correction
ANOVA:
Here is our coding for our inputs for delta POS:
15
Similarly, here is the coding for delta ORI:
16
The coding for both, delta POS and delta ORI, is:
The summary of coding is below:
Direction
Coding
Y+ 1Y- 2X+ 3
TypeCoding
UNI 1INT 2
This will come in handy later, when we look at profile plots.
17
In SPSS we use Analyze>Mixed Models>Linear to run our two-way repeated measures analyses.
We do this twice, for delta POS and delta ORI.
For delta POS, we first check to see which covariance structure is the best for the data. We compared AR(1) and Compound Symmetry (CS). AR(1) had the lowest AICC (Hurvich and Tsai’s Criterion). Therefore, we use the ANOVA results generated using the AR(1) covariance structure.
Model Dimensiona
Number of Levels
Covariance Structure
Number of Parameters
Subject Variables
Number of Subjects
Fixed Effects
Intercept 1 1Direction 3 2Type 2 1Direction * Type
6 2
Repeated EffectsType * Direction
6 First-Order Autoregressive
2 Subject 8
Total 18 8a. Dependent Variable: Delta POS.
This is a quality control check on our data entry. There are 3 directions, and 2 types of movements. Also, there are 8 subjects.
18
Information Criteriaa
-2 Restricted Log Likelihood
-6.762
Akaike's Information Criterion (AIC)
-2.762
Hurvich and Tsai's Criterion (AICC)
-2.455
Bozdogan's Criterion (CAIC)
2.713
Schwarz's Bayesian Criterion (BIC)
.713
The information criteria are displayed in smaller-is-better forms.
a. Dependent Variable: Delta POS.
Here, we can see that the AICC is -2.455.
19
From here, we can see that Direction and Type are significant variables, with p-values at 0.005 and <0.0001, respectively. The interaction variable, Direction * Type, is not significant, with a p-value of 0.829.
Estimates of Covariance Parametersa
Parameter Estimate Std. Error
Repeated Measures
AR1 diagonal
.046877 .012366
AR1 rho .496365 .135037a. Dependent Variable: Delta POS.
From here we can see that the estimated correlation between measurements taken on subjects 1 and 3, is the same between subjects 1 and 8, namely (0.496365)^2 = 0.25
We do the same for delta ORI, and we find that in this case, CS is the better covariance structure compared to AR(1). Here are our results:
20
Type III Tests of Fixed Effectsa
Source Numerator df
Denominator df
F Sig.
Intercept 1 10.042 130.175 .000Direction 2 31.657 6.182 .005Type 1 30.842 18.281 .000
Direction * Type
2 33.232 .189 .829
a. Dependent Variable: Delta POS.
Model Dimensiona
Number of Levels
Covariance Structure
Number of Parameters
Subject Variables
Number of Subjects
Fixed Effects
Intercept 1 1Direction 3 2Type 2 1Direction * Type 6 2
Repeated EffectsType_again * Direction_again
6 Compound Symmetry
2 Subject 8
Total 18 8a. Dependent Variable: Delta ORI.
This is our quality control check on our data entry. Again, we have the right values of Direction (3), Type (2), and Subject (8).
Information Criteriaa
-2 Restricted Log Likelihood
-3.122
Akaike's Information Criterion (AIC)
.878
Hurvich and Tsai's Criterion (AICC)
1.185
Bozdogan's Criterion (CAIC)
6.353
Schwarz's Bayesian Criterion (BIC)
4.353
The information criteria are displayed in smaller-is-better forms.a. Dependent Variable: Delta ORI.
Our AICC value is at 1.185. This is lower than the AICC value for AR(1) (not shown here).
Type III Tests of Fixed Effectsa
21
Source Numerator df
Denominator df
F Sig.
Intercept 1 7 121.802 .000Direction 2 35.000 2.314 .114Type 1 35.000 .874 .356Direction * Type
2 35.000 3.574 .039
a. Dependent Variable: Delta ORI
Here we see that the interaction effect is significant at a p-value of 0.039. The first step in a two-
factor ANOVA is to look at the interaction effect. If it is significant, we cannot interpret the main
effects, in this case Direction and Type. This interaction effect tells us that the failure of the delta
ORI to one factor, say Direction, is the same for different levels of the other factor, Type. Since
the interaction term is multiplicative, it can have a “large and important impact” on the delta
ORI.
Estimates of Covariance Parametersa
Parameter Estimate Std. Error
Repeated Measures
CS diagonal offset
.032867 .007857
CS covariance .013377 .010163a. Dependent Variable: Delta ORI.
22
We can also create profile plots in SPSS that will show us the trend of delta POS as it relates to direction of movement and type of movement.
It is handy to have the coding here, from page 18:
Direction
Coding
Y+ 1Y- 2X+ 3
TypeCoding
UNI 1INT 2
23
Here, we see that as direction moves from 1 to 3, that is, from Y+ to Y-, to X+, the mean of delta POS drops.
24
Here, we see a profile plot for type of movement:
Here, we can see that the mean of delta POS is lower at Type = 1, which is Unintentional, than at Type = 2, which is Intentional.
25
We did not create profile plots for delta ORI since only the interaction term is significant.
Inferential Statistics and Conclusions:
Movement type and direction of movement significantly affect delta POS. Mean of delta POS drops as direction goes from Y+ to Y- to X+. As movement type goes from Unintentional to Intentional, the mean of delta POS rises. The interaction term is not significant.
The mean of delta ORI is affected by the interaction between direction of movement and movement type. Interaction means a difference in the response at one level of one factor to be different at different levels of another factor.
Again, it is important to note here that the results of this study cannot be used for inference for any population but the eight subjects used in this study, under the same conditions under which they were tested. This is due to the non-random nature of the sample.
26
References
STAT 502 Notes. https://onlinecourses.science.psu.edu/stat502/node/189
Green, Samuel B., and Neil J. Salkind. Using SPSS for Windows and Macintosh: analyzing and understanding data . 4th ed. Upper Saddle River, NJ: Pearson/Prentice Hall, 2005. Print.
27