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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.

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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

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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:

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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.

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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:

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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:

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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


Lower Bound

.4069

Upper Bound

.6794

5% Trimmed Mean .5387

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Median .5795Variance .065Std. Deviation .25565Minimum .16Maximum 1.01Range .85Interquartile Range .49Skewness .167 .564Kurtosis -1.095 1.091

Y+

Mean .6499 .06888


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:

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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



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:

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DescriptivesType of Movement Statistic Std.

Error

Delta POS

INT

Mean .6978 .05476


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


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.

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And finally, for delta POS, our tests for normality indicate that normality has not been violated:

Tests of NormalityType of Movement


Delta POSINT .116 24 .200* .958 24 .397UNI .124 24 .200* .955 24 .354


For delta ORI:

Case Processing SummaryDirection of Movement



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


Lower Bound

.5022

Upper Bound

.7215

5% Trimmed Mean .6174

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Median .6315Variance .042Std. Deviation .20577Minimum .23Maximum .90Range .67Interquartile Range .34Skewness -.271 .564Kurtosis -1.031 1.091

Y-

Mean .5182 .05849


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


Lower Bound

.3582

Upper Bound

.5965

5% Trimmed Mean .4713Median .4862Variance .050Std. Deviation .22356Minimum .12Maximum .94Range .82

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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


Delta ORI

X+ .145 16 .200* .950 16 .482Y- .154 16 .200* .943 16 .385Y+ .098 16 .200* .977 16 .939


Now, we do our quality control check for type of movement:

Case Processing SummaryType of Movement


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

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Type of Movement Statistic Std. Error

Delta ORI

INT

Mean .5113 .04034


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


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.

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And finally, our tests for normality tell us that this assumption is not violated:

Tests of NormalityType of Movement


Delta ORI

INT .133 24 .200* .949 24 .253UNI .109 24 .200* .954 24 .327


ANOVA:

Here is our coding for our inputs for delta POS:

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Similarly, here is the coding for delta ORI:

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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.

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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.

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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.

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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:

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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

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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.

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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

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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.

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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.

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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.

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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.

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http://www.bibme.org/

http://www.bibme.org/

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storage.googleapis.com › wzukusers › user-3393… · Web viewMost applied statisticians choose the multivariate test over the others. In SPSS using this procedure, difference