Magnets and Pain Relief
Web Resource: Rice Virtual
Lab in Statistics By David M. Lane
Case:
Magnets and Pain Relief
Background, Experimental Design, and Materials
Click the above link and a new web page, Magnets
and Pain Relief, will appear. Next, choose Background, Experimental design, and
Materials from the list to the left.
Can the application of
magnetic fields be an effective treatment for pain?
Variables Involved
______________________________________________
One Independent Variable with
Two Levels
Device: an active magnetic
device vs. an inactive magnetic device
Patients experiencing
post-polio pain syndrome were recruited. Half of the patients were treated
with an active magnetic device and half were treated with an inactive device.
Coding: The subjects in the
experimental group were treated with an active magnetic device and they were
coded as 1. The subjects in the control group were treated with an inactive
placebo and they were coded as 2.
Dependent Measures: Pre and
Post treatment Scores
All subjects grades pain at
the trigger point under palpitation on a scale from 1 to 10 (1 is the least
pain, increasing to 10). Pain was graded before and after application of the
above device.
Data Analysis
Simplified Presentation
To simplify the presentation, an independent
t-test will be introduced here. The independent variable is the device with
two levels: an active magnet and an inactive placebo. The dependent variable
is the post-treatment scores, score_2.
The Null hypothesis
There is no difference in post-treatment ratings
of pain between the two groups.
The Alternative Hypothesis
The mean rating of pain for the experimental
group is lower than that for the control group. The
application of magnetic fields is an effective treatment for pain. It
is a one-sided test.
Set a
Significance Level
Use a .05 significant level.
Assumptions
An independent t-test makes three assumptions:
-
The populations are normally distributed.
-
The variances in the populations are equal.
-
Each observation is sampled randomly and is
independent of each other observation.
If the assumptions
are violated, the test results may not be valid.
Obtain the Raw Data
Copy the Data Set
Click the Raw Data section.
It is listed on the left side of the web page: Magnets and Pain Relief.
Highlight the variable names and all the scores. Then right click on
the highlighted area and select Copy from the pop-out menu.
Now you may minimize or close the Magnets
and Pain Relief window.
Start a Word Processing
Program
For example, if you have the Microsoft Word
program, you may start Microsoft Word by choosing Start \ Programs \ Microsoft
Word.
Paste Raw Data
Choose Edit \ Paste. The raw data set will
appear.
Save the File
To save the file, choose File \ Save as.
Save in: Select a preferred drive.
Save as type: click the down arrow and select Text only (*.txt)
File name: Score.txt.
Click save. Click Yes.
Exit Microsoft Word.
Import the File to SPSS
Start the SPSS. Choose File \ Read Text Data. The
Open File dialog will appear. Select the drive and the file. Click Open. The
Text Import Wizard window will appear.
Step 1: Does your text
file match a predefined format?
No. The first line in our text file is variable
names. Click the Next button.
Step 2: How are your
variables arranged?
Delimited. Spaces are used to separate variables
in our text file.
Are variable names included at the top of your
file?
Click "Yes". Click the Next button.
Step 3: The default
selections are correct.
Click the Next button.
Step 4: Which delimiters
appear between variables?
Space. Click the Next button.
Step 5: Specifications
for variable(s) selected in the data preview.
Four Variables
The variable "score_1" was the pain rating
measured before the treatment.
The variable "score_2" was the pain rating measured after the treatment.
The variable "change" is the difference between the two ratings (score_1 -
score_2 = change).
The variable "active" specifies the group membership. The experimental group
was coded as 1 and the control group was coded as 2.
Click each variable in the Data preview window.
The variable name and data format will display
in the text box.
Click the Next button.
Step 6: Click the Finish
button. The data set will appear in the SPSS Data Editor window.
Imported Variables
The variable "score_1" was the pain rating
measured before the treatment.
The variable "score_2" was the pain rating measured after the treatment.
The variable "change" is the difference between the two ratings (score_1 -
score_2 = change).
The variable "active" specifies the group membership.
The experiment group was coded as 1 and the control group was coded as
2.
Part One
Screen Data
Descriptive Statistics
Choose Analyze \ Descriptive Statistics \
Explore. First, move the variable "score_2" to the Dependent List. Next, move
the variable "active" to Factor List.
The Statistics Button and the Plots button: Click
the Statistics button and select Outliers and click Continue. Click the Plots
button and select Histogram and Normality plots with tests. In the Spread vs.
Level with Levene Test, select Power estimation.
What does power estimation mean? Right click on
Power estimation. The definition will pop out.
Finally, click continue and OK.
Results
1. Number of Subjects
The number of subjects in the experimental group
is 29. The number of subject in the control group is 21.
2. Descriptive
Statistics for Each Group
Examine measures of central tendency and measures
of variability.
Subjects grade pain at the trigger point under
palpitation on a scale from 1 to 10 (1 is the least pain, increasing to 10).
Mean
The mean pain rating for the experimental group
is 4.379. The mean pain rating for the control group is 8.429. It appears
that the treatment group experienced less pain.
Standard Deviation
The sample standard deviation for the
experimental group is 3.144. The sample standard deviation for the control
group is 1.859. It appears that the pain ratings are more variable in the
experimental group.
Deviate from Normality
A value of
zero for the
skewness indicates a symmetric distribution. A value of
zero for the
kurtosis indicates the a shape (flatness or peakedness) close to normal. The
normal distribution has kurtosis of zero.
Experimental Group
Skewness = .541 and Standard Error = .434
The value of the coefficient of skewness is
slightly more than 1 times its standard error. For our example, .541 /
.434 = 1.25
Kurtosis = -.546 and Standard Error = .845
For our example, -.546/.845 = -.646. The value of the coefficient of
kurtosis is less than 1 times its standard error.
We will visualize the distribution by plotting a histogram and a boxplot.
Control Group
Skewness = -1.062 and Standard Error = .501
For our example, -1.062/.501 = -2.12. The
value of the coefficient of skewness is more than 2
times its standard error. The distribution is
negatively skewed.
Kurtosis = .175 and Standard Error = .972
For our example, .175/.972 = .18. The value of the coefficient of kurtosis
is less than 1 times its standard error.
We will visualize the distribution by
plotting a histogram and a boxplot.
Interval Estimation
(1) Experimental Group: Based on the sample mean and standard error of the
mean, a confidence interval can be computed. The 95% confidence interval for
the mean is 3.183 to 5.575.
(2) Control Group: The 95% confidence interval for the mean is 7.582 to
9.275.
(3) Note that the two confidence intervals do not overlap. A significant
mean difference will be found between the two groups.
3. Test of Normality
The hypothesis of normality of post-treatment
scores for both groups was rejected, p < .05.
4. Test of Homogeneity of
Variance
Based on Mean
The null hypothesis that the two group variances
are equal was rejected, p < .05.
5. Histograms
Experimental group
Control Group
It appears that the distribution of the scores
for the control group is negatively skewed. More subjects in the control group
reported higher degrees of pain.
6. Normal Q-Q Plots
If the sample is from
a normal population, the data points should fall on a straight line.
Experimental group
Control group
The normal Q-Q plots showed that the values
deviated somewhat from the straight line.
7. Detrended Normal Q-Q
Plots
If the sample is from
a normal population, the data points are expected to cluster around a
horizontal line through 0. Also, there should be no pattern.
Experimental group
Control group
Note that the
deviations from a straight line are not randomly distributed around 0.
8. Boxplots
The side-by-side boxplots for the two groups on
the dependent variable, score_2 (post-treatment ratings of pain), is displayed
below
Variability
The length of the box
represents the difference between the 25th and 75th percentiles.
The larger the box, the greater the spread of the
data.
Note
that the pain ratings are more variable in the experimental
group.
Skewness
1. Median
If the median is not in
the center of the box, the distribution is skewed.
If the median is closer to the top of the box,
the distribution may be negatively skewed. If the median is closer to the
bottom of the box, the distribution may be positively skewed.
2. Length of
Whiskers
Draw lines from the
ends of the box to the largest and smallest values that are not outliers.
These lines are called whiskers. If the upper whisker is much longer
than the lower whisker, it gives the impression of positive skewness. If the
lower whisker is much longer than the upper whisker, it gives the impression
of negative skewness. Note that your data set should be large enough to
make the above observation.
Outliers
Case numbers are
used to label outliers (o) and extremes (*). The outliers are cases with
the values between 1.5 and 3 box-lengths from the 75th percentile or 25th
percentile. The extreme values are cases with the values more than 3
box-lengths from the 75th percentile or 25th percentile.
Note that there are two outliers detected in the control group. The outliers
may be due to recoding errors, due to the sample being from a skewed
population distribution or not being from the same population, or simply due
to the small sample size.
Should you delete the outlier? Look for a reason why it happened. Once you
know the reason, the further course of analysis becomes obvious.
9. Spread-versus-Level
Plot
The spread-versus-level plot plots the nature
logs of the interquartile ranges against the natural logs of the medians for
each group. Note that there is a strong negative relationship (slope = -.725)
between the spread and level. A power transformation can be used to lessen the
relationship.
power = 1 - slope = 1.725
Thus, the closest power is 2 (square).
The following commonly used transformations are
listed in the SPSS Base System User's Guide.
|
Power |
Transformations |
|
3 |
Cube |
|
2 |
Square |
|
1 |
No Change |
|
1/2 |
Square Root |
|
0 |
Logarithm |
|
-1/2 |
Reciprocal of the square root |
|
-1 |
Reciprocal |
Remedy
We have detected that our data violate
assumptions of normality and homoscedasticity. In this case, transforming the
data or use a nonparametric test may provide a better analysis.
Ask Why
When an assumption is violated, the correct
course of action is to find a reason why it happened. Once you know the
reason, the further course of analysis becomes obvious.
Normalize
the Skewed Distribution
If the coefficient of
skewness is not statistically significant, the departure of the distribution
from normality can be considered due to random factors, and the distribution
can be normalized. In general, area transformations are a better method of
the normalization of data than other methods, as, e.g., the square root
method, the logarithm transformation, or the often used arc sine
transformation.
If the coefficient of
skewness is statistically significant, other avenues leading to normality
should be explored. Normalizing markedly skewed distributions may obscure
factors making the distribution skewed to begin with. These factors may, in
some cases, be of crucial importance.
Rank the Data
If the distribution does not
appear to be normal and the sample size is small, we may consider
statistical procedures that do not
require the assumption of normality (distribution-free or nonparametric
tests) and transform the interval or ratio data to the ordinal data. The
scores will be ranked from smallest to largest values.
The Mann-Whitney test (Wilcoxon
Rank-Sum) is a non-parametric analog of the independent t-test. However the
t-test procedure will always have more power than the corresponding
non-parametric test if the distribution is normal.
Square Transformations
Suppose that the researcher decided to transform
the data to stabilize the variances.
A power transformation is often used to stabilize
variances (SPSS Base System User's Guide, Release 6.0).
First, the researcher would like to to correct
nonnormality and unequal variances by transforming all the data values of the
variable "score_2".
Transformations are usually chosen from the
"power family" of transformations. To achieve equal variances in the groups, a
square transformation is suggested by the SPSS. Switch to the Data Editor
window. To apply the square transformation, choose Transformation \ Compute.
Type square in the Target Variable text box.
Next, type score_2, *, and score_2 in the Numeric
Expression text box.
Click OK. The new variable, square, will appear
in the Data Editor window.
To judge the success of the transformation,
choose Analyze \ Descriptive Statistics \ Explore. Move the previously
selected variable, score_2, back to the variable list. Click the new variable,
square, and move it to the Dependent List. Click OK.
Results
1. Examine descriptive
statistics
Standard Deviation
The sample standard deviation for the
experimental group is 34.05. The sample standard deviation for the control
group is 28.20. The standard deviations for the two groups are more equivalent
compared to the original, untransformed data. The square transformation did
help.
Skewness and Kurtosis
The values of skewness and kurtosis should be
close to zero after a successful transformation.
For our example, the skewness for the
experimental group is increased to 1.363 (about 3 times its standard error).
The skewness for the placebo (control) group is decreased to -.736 (about 1
times its standard error).
For the experimental group, the value of kurtosis is less than 1 times its
standard error (.477/.845 = .56). For the control group, the value of kurtosis
is less than 1 times its standard error (-.712/.972 = -.73).
What do you conclude?
2. Test of Normality
The hypothesis of normality of post-treatment
scores for both groups was still rejected, p < .05.
3. Test of Homogeneity
of Variance
The null hypothesis that the two group variances
are equal was not rejected, p > .05.
A power transformation is often used to stabilize variances (SPSS Base System
User's Guide, Release 6.0). A square transformation achieved equal variances
in the groups.
4. Data Visualizations
The square transformation was not successful in
normalizing the skewed distributions. Observe the following graphs.
Histograms
Normal Q-Q Plots
If
the sample is from a normal population, the data points should fall on a
straight line.
Experimental Group
Control Group
Detrended Normal Q-Q
Plots
If the sample is from a normal
population, the data points are expected to cluster around a horizontal line
through 0. Also, there should be no pattern.
Experimental group
Control group
Boxplot
Variability
The length of the box
represents the difference between the 25th and 75th percentiles.
The larger the box, the greater the spread of the
data. Note that the variability of
pain ratings are similar for the two groups.
Length of Whiskers
Draw lines from the
ends of the box to the largest and smallest values that are not outliers.
These lines are called whiskers. If the upper whisker is much longer
than the lower whisker, it gives the impression of positive skewness. If the
lower whisker is much longer than the upper whisker, it gives the impression
of negative skewness.
Conclusion
A square transformation did achieve equal
variances in the groups. However, the square transformation was not successful
in normalizing the skewed distributions.
Part Two
Inferential Statistics
Conduct the Mann-Whitney test.
The square transformation was not successful in
correcting the nonnormality. The researcher then decided to use a
nonparametric test.
If the distribution does not appear to be
normal and the sample size is small, we may consider statistical procedures
that do not require the assumption of normality (distribution-free
or nonparametric tests) and transform the interval or ratio data to the
ordinal data. The scores will be ranked
from smallest to largest values. The Mann-Whitney test (Wilcoxon Rank-Sum)
is a non-parametric analog of the independent t-test. (The Kruskal-Wallis
method is a non-parametric analog of one-way ANOVA.) However the
t-test/ANOVA procedures will always have more power than the corresponding
non-parametric tests if the distribution is normal.
Choose Analyze \ Nonparametric Tests \ Two
Independent Samples. Move the variable "score_2" to the test Variable list.
Move the variable "active" to the Grouping Variable list. Click on the define
Groups button. Group1: Type 1 in the textbox. Group 2: Type 2 in the textbox.
Click Continue.
Click the Options button and select Descriptive
statistics. Click Continue and OK.
Results
It begins by combining the scores from two groups
into a single set. These scores are then rank-ordered from lowest to highest.
Mean Rank
The mean rank for the experimental group is lower (18.31) than the mean rank
for the control group (35.43).
Sum of Ranks
If the two populations have the same
distribution, their sample distributions of ranks should be similar.
Note that the sum of the ranks for the
experimental group (531) was much smaller than the control group (744).
Test Statistics
Wilcoxon W
The number (531) identified as
W represents the sum of the ranks for the
experimental group. If the two populations have the same distribution, their
sample distributions of ranks should be similar (SPSS Base System User's
Guide).
Recall that the sum of the ranks for the experimental group (531) was much
smaller than the control group (744).
Mann-Whitney U
The number (96) identified as
U represents the number of times a value
in the experimental group precedes a value in the control group. If the two
populations have the same distribution, values from one group should not
consistently precede values in the other (SPSS Base System User's Guide).
The Z value and the Associated Probability
The z value was -4.418. The probability
associated with a z value of -4.418 was was less than .05. The null hypothesis
was rejected. The two groups were significantly different. The experimental
group experienced less pain.
Conduct an independent t test.
Parametric techniques are preferred because of
their greater sensitivity. That is, they are more likely to
lead to reject the null hypothesis.
Test the consequences of assumption violation by
Simulation
Open the web page,
http://www.ruf.rice.edu/~lane/case_studies/magnets/index.html, click the
Inferential Statistics section. Dr. David Lane suggested to run the "Data
Analysis Lab" and to test the consequences of these assumption violations by
simulation. The results indicates a significant difference can be used to reject
the null hypothesis even with the assumption violation.
Conduct an independent t test. The dependent
variable will be the post-treatment measure, score_2 and the independent
variable will be the variable "active".
The Null hypothesis
There is no difference in post-treatment ratings
of pain between the two groups.
The Alternative Hypothesis
The mean rating of pain for the experimental
group is lower than that for the control group. The
application of magnetic fields is an effective treatment for pain.
It is a one-sided test.
Choose Analyze \ Compare Means \ Independent
Samples T Test. Move the variable "score_2" to the test Variable list. Move the
variable "active" to the Grouping Variable list.
Click on the define Groups button. Group1: Type 1
in the textbox. Group 2: Type 2 in the textbox. Click Continue.
Click OK.
Results
Group Statistics
The group treated with the active magnet reported
lower pain ratings (M = 4.38, s = 3.14) than did the group
treated with the inactive placebo (M = 8.429, s = 1.86). Note
that the mean difference is very large (4.379 - 8.429 = -4.05) and the two
standard deviations are not equal (3.14 vs. 1.86).
Independent Samples Test
Equality of Variances
Levene's test is used to test the null
hypothesis that the two population variances are equal.
The probability
associated with the F test is less than .05. The null hypothesis was
rejected, F = 4.458, p = .04. The assumption of
homoscedasticity was not met.
Equal Variance Not
assumed
Examine the t ratio
corrected for inequality of variance (the Welch
t test)
The probability associated with the t
value was less than .05. The null hypothesis was rejected.
The group treated with the active magnet reported
significantly lower pain ratings (M = 4.38, s = 3.14) than did
the group treated with the inactive placebo (M = 8.429, s =
1.86), t(46.418) = -5.695, p < .05.
95% Confidence Interval of the difference
The 95% confidence interval for the difference
between two population means was from -5.48 to -2.6184.
Test hypotheses by using confidence interval
of the difference
Note that the null hypothesis of no difference
was rejected and the 95 % confidence interval for the difference
between two population means did not include 0 (the null value). The
null hypothesis value falls outside the 95% confidence interval around the
sample mean difference.
Part
Three Q and A
Open the web page:
Magnets and Pain Relief. Choose Descriptive Statistics, Inferential
Statistics, and Interpretation from the list to the left and answer the
associated questions.
Web Resources
http://www.davidmlane.com/hyperstat/dist_free.html by David Lane