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					                       BFS Statistical Analysis

                    A Statistical Analysis of Benign
                    Fasciculation Syndrome (BFS)



                             Patrick Bohan

                              PO Box 331

                            109 Raven Way

                     Buena Vista, Colorado, USA

                            719-966-5167

                         pbohan1@gmail.com

Abstract:



Defining and understanding neurological disorders has been a medical

mystery. Benign Fasciculation Syndrome (BFS) is one such disorder.

BFS is sometimes referred to as Peripheral Nerve Hyperexcitability

(PNH). BFS or PNH is a neurological disorder and its cause is not

entirely understood, but it theorized that the cause may stem from an

imbalance between potassium and sodium at the nerve endings. This

imbalance is what causes involuntary impulses that consequently

stimulate the nerve endings causing them to fire and twitch. Other BFS

symptoms include muscle fatigue, cramps, pins and needles, muscle

vibrations, headaches, itching, sensitivity to temperatures, numbness,

muscle stiffness, muscle soreness and pain. Like most neurological

disorders, there is no cure for BFS. The purpose of this writing is to



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                       BFS Statistical Analysis

better define and understand the relationship between BFS symptoms,

body parts affected by BFS, the potential causes of BFS, and potential

remedies for BFS. To accomplish this task, a survey was conducted

and data was obtained from 125 people who have been diagnosed with

BFS or have BFS like symptoms. The data was analyzed using a simple

statistical analysis to find the mean, median, mode, standard

deviation, variance, range, percentile rank, skewness, standard error,

and coefficient of variance for each symptom, body part affected, and

potential remedy. The data was also modeled using a linear regression

analysis to determine if there is correlation between symptoms,

potential causes or triggers, body parts affected by BFS, and potential

remedies.


Sources of Information:


Since there are few medical publications on BFS, I rely on information

posted by fellow BFS sufferers on two user sites. I only used

information in this writing that has been corroborated by a multitude

of BFS sufferers. After all, the people suffering from BFS are obviously

the experts on the syndrome. The internet user sites are listed below:


Facebook: https://www.facebook.com/#!/groups/88467288815/

Internet: http://www.nextination.com/aboutbfs/




                                    2
                       BFS Statistical Analysis

Background:



BFS sufferers live in fear because similar symptoms can be found in

other crippling and deadly disorders such as Parkinson Disease,

Amyotrophic Lateral Sclerosis (ALS), Multiple Sclerosis (MS), and even

brain tumors. Because of this, many BFS patients have been forced to

undergo advanced medical testing including Magnetic Resonance

Imaging (MRI) performed on the brain as well as an Electromyography

(EMG) to rule out other neurological disorders. Anyone with BFS, or

doctors that have studied BFS, will tell you that “benign” is a bad word

to describe the disorder. People may not die from BFS, but it can be

debilitating. In fact, many BFS sufferers have similar symptoms to

other neurological disorders including Nueromyotonia (NMT), Benign

Cramp Fasciculation Syndrome (BCFS), fibromyalgia, Reflex

Sympathetic Dystrophy (RSD), stiff person syndrome, continuous

muscle fiber activity, continuous motor nerve discharges, and Isaac

Syndrome. Many remedies attempted to relieve BFS symptoms are

exactly the same as those remedies used for NMT, BCFS, RSD and

other neurological disorders. At this time there is no evidence that BFS

sufferers are any more likely to acquire other more serious

neurological disorders, such as ALS or MS, than any “normal” person.


Purpose:


                                   3
                        BFS Statistical Analysis

Most BFS sufferers have been to multiple general practitioners and

neurologists looking for answers but have failed to receive any logical

explanations. Since BFS is benign there are no or very few studies on

the disorder and therefore, doctors do not have any answers. After

each doctor visit the medical files of BFS sufferers are put in a file

cabinet and locked away. How is this going to find a cure for BFS? It

will not! I have also come to realization that doctors are not

necessarily the best mathematicians to find solutions by comparing

and reviewing data (conversely, I do not understand medicine as well

as doctors). In fact, since doctors do not compare the medical records

of people with similar ailments (I am not blaming doctors because I

realize they may not have the tools to accomplish this task), they treat

each patient like a guinea pig using a trial error approach to find a

drug regimen that may work to alleviate some symptoms. And yes,

what works for one person afflicted with BFS may not necessarily work

for another person afflicted with BFS, so it is hard to pin point a

treatment regimen for BFS sufferers. If, on the other hand, doctors

were supplied the results of this study, they would better understand a

starting point to treat their patients. For instance, the results indicate

that people whose BFS symptoms get worse due to a sickness will

have more success using benzodiazepine drugs to alleviate symptoms

than other drug classifications. However, benzodiazepine drugs are not


                                     4
                       BFS Statistical Analysis

as helpful in patients who believe their symptoms get worse due to

stress – anti-convulsants may work better. We live in a verbal society,

but numeric analysis is needed to solve the complex problems and

mysteries of life. Having doctors understand these differences for

treating BFS would be one purpose for writing this paper.


Therefore, I created a survey to anonymously obtain the medical

records of BFS sufferers into one location so we can statistically

analyze the data to better define and understand the ailment. I

completely understand that scientists, doctors, and researchers are

spending most of their time trying to solve Parkinson’s disease, ALS,

and MS since these diseases are, without question, much worse. Once

there are cures for these diseases, then it is possible that cures for

BFS could follow shortly afterwards. However, it is debatable as to

whether or not this approach to solving the mysteries of neurological

disorders is the best or most logical. As an engineer, I saw many

projects fail because we tried to design products that incorporated too

many features. This created many design complications and ultimately

these projects failed. On the other hand, multiple products that

focused on particular features were more successful, and over the

course of time the features can eventually be incorporated into one

product (for example - the phone camera). This approach to problem

solving saved the company both in cost and time to market. The same

                                    5
                       BFS Statistical Analysis

can be said of medicine – maybe it makes more sense to focus on less

complicated disorders such as BFS or RSD and apply what is learned to

more complicated neurological disorders such as ALS and MS. This

seems to be a fundamental issue when trying to solve problems (in my

opinion) – everyone wants to hit a home run instead of making small

incremental advances, regardless of the profession. Hopefully, the

analysis included in this paper will provide one of those small

incremental advances in not only understanding BFS, but the

mysteries of all neurological disorders.


The Survey:


A survey was created in Google Docs and can be found at the following

link:

https://spreadsheets.google.com/spreadsheet/viewform?hl=en_US&au

thkey=CJvBgaQM&formkey=dElCQkFBRWlvY1ZSTThKTmNsbEg4d0E6M

Q#gid=0


I will keep the survey open indefinitely with the hope that we can

continue to grow the sample size and therefore, better understand the

disorder. I will periodically update the data on my website (links to

specific types of data are listed throughout this paper).




                                    6
                           BFS Statistical Analysis

The Survey can also be reached from my BFS webpage:

http://patrickbohan.home.bresnan.net/BFS.htm. Click on the link “BFS

Survey”.


Data:


The excel data file for all 125 responses can be found on my BFS

website: http://patrickbohan.home.bresnan.net/BFS.htm. Click on the

link “Survey Data” and open the first tab titled “BFS”. This is the data

file that will be statistically analyzed except when remedy or treatment

variables are being analyzed. I use the data on the “BFS No Zero” tab

to statistically analyze remedy or treatment variables (this will be

explained in this text).


Data points with brackets “[]” around them were identified as outliers

because these responses were outside plus or minus 3 standard

deviations from the mean for all tested variables. Most outliers were

determined from the parameters: Symptom Averages, Body Part

Averages, and or Remedy Averages. A handful of other outliers were

determined by running a statistical analysis on each variable. Outliers

are omitted from any statistical analysis.


Data Summary:




                                      7
                        BFS Statistical Analysis

The statistical analysis data for each parameter can be found at:

http://patrickbohan.home.bresnan.net/BFS.htm. Click on the link

“Survey Data” and view the excel file tab titled “Data Summary” to

find a statistical summary of all parameters in the survey. The tab

“Calculate” contains the averages for all parameters in the survey.


A lot of the statistics on the “Data Summary” tab are irrelevant. For

instance, statistical data for variables that had yes or no responses (1

or 0 answers respectively) are for the most part irrelevant. Variables

such as EMG, MRI, Sickness, Flu Shot, Chemicals, Exercise, Altitude,

Stress, History, Spine Injury, Sex, Remedies, and Missing had yes / no

responses – meaning other than the statistical average, most of the

other statistical results have very little meaning. Even statistical

results for variables that had multiple response options such as

variables Region or Day are for the most part irrelevant. More relevant

results for these parameters can be found on the “Calculate” tab,

which merely computes statistical averages. On the “Calculate” tab the

results to these questions are sorted to determine for instance, how

many people in the survey where from Europe or North America. The

“Calculate” tab results are shown below in Table I below (the

classification of variables, ie General (G), will be defined later and are

color coded on the “Data Summary” and “Calculate” tabs):




                                     8
                     BFS Statistical Analysis

Table I: ―Calculate‖ Tab Results


__________________________________________________

__________________________________________________

___________________


General (G):

Age: 38.74

Sex: 64.5% Male

Region: 68% North America; 1.6% South America; 26.4% Europe;

0.8% Europe; 3.2% Oceania

MRI: 56.8% Yes

EMG: 66.9% Yes

Years with symptoms: 3.49 years

Years diagnosed: 2.23 years



Causes / Triggers (CA):

Flu shot: 10.4%

Chemicals: 4.8%

Prescription drugs: 20.8%

Neck/Spine injury: 13.6%

Sickness: 28.8%

Exercise: 20%



                                  9
                         BFS Statistical Analysis

Stress: 72.6%

History: 19.2%

Other: 20%



The sum of causes adds to more than 100% because people selected

multiple potential causes (this is okay).



Stressers (ST):

Sickness 3.86 (out of 10)

Exercise: 5.59

Stress: 6.83



Symptoms (S):

Twitching: 7.64 (out of 10)

Pins and Needles: 3.72

Cramps: 3.34

Muscle Fatigue: 3.97

Headaches: 2.85

Itching: 2.13

Numbness: 2.79

Muscle Stiffness: 3.98

Muscle Vibrations/Buzzing: 4.7


                                    10
                         BFS Statistical Analysis

Muscle Pain/Soreness: 4.52

Sensitivity to Temperature: 3.06

Symptom Average: 3.87



Body Part (B):

Feet: 5.68 (out of 10)

Lower Leg: 7.28

Upper Leg: 5.15

Hip/Butt: 3.83

Back: 3.29

Abdomen: 2.88

Chest: 2.43

Head/Neck: 3.53

Hands: 4.4

Arms / Shoulders: 4.83

Body Average: 4.32



Remedies (RE):

Benzodiazepine: 3.91 (out of 10, for those that tried the treatment);

54.5% did not try the method

Anti-Convulsant: 2.56; 57.7%

Anti-Depressant: 2.11; 54.1%


                                    11
                      BFS Statistical Analysis

Potassium Channel: 1.4; 87.8%

Sleeping Pills: 3.02; 66.7%

Muscle Relaxant: 2.2; 66.4%

Homeopathic: 2.2; 63.4%

Supplements: 2.27; 25.2%

Diet: 2.03; 48%

Acupuncture: 2.14; 82.1%

Massage: 2.35; 50%

Yoga: 2.46; 71.5%

Remedy Average: 2.39; 6.5%



Various (V):

Time: 4.91 (People feel their symptoms are slightly improving over

time since a 5 means that symptoms have stayed the same)

Day: 32% Morning; 29.6% Day/Evening; 38.4% Night (Time when

symptoms are worse)

Remedies: 15.2% of people said that certain remedy treatments made

their symptoms worse.

Missing: 9.6% of people said that a remedy solution that worked for

them was not included in the survey.

Altitude: 5.6% of people said their symptoms got worse at altitude.




                                  12
                       BFS Statistical Analysis

________________________________________________________

________________________________________________________

__________________



However, for many parameters, just knowing the mean (average)

does not really describe the variable without knowing more about the

result such as its standard deviation. For instance, if a child scored 5

points below the average on a test, this does not tell us much

statistically without understanding how the class did as a whole. If the

child’s test result was within one standard deviation of the class

average, than the child’s result would still rank in the middle of the

class (a C grade – see Figure 1). If, on the other hand, the result was

over 2 standard deviations away from the mean, than the child’s result

would rank in the bottom of the class (a D or F grade). Thus,

understanding the standard deviation, variance, and standard error of

the class distribution would be extremely helpful.


Let’s examine the results of one parameter on the “Data Summary”

tab, Twitching (Figures 2 and 3 below summarize the data results for

the variable twitching). The question in the survey for the variable

twitching specifically reads “Enter a number from 1 to 10 on how much

the symptom twitching affects you? A 1 means the symptom does not

affect you at all, and a 10 means the symptom occurs 24/7”. For the

                                    13
                          BFS Statistical Analysis

rest of this writing I will refer to this question simply as “twitching”.

The results of the variable twitching are summarized below in Table II:



Table II: ―Data Summary‖ Tab Results for Twitching

__________________________________________________

__________________________________________________

___________________


                      i
   Mean – 7.64


   The mean is the arithmetic average. In the case of twitching the

   mean is 7.64. Hence, BFS sufferers, on average, feel twitching in

   their body’s 76.4% of the time. The mean is also illustrated in

   Figure 2; it is the point at which the red bell cure line is at its

   maximum point.


                 ii
   Median – 8


   The median is the result at which 50 percent of the survey

   responses are above the result and 50 percent are below the result.

   In the case of twitching the median result is 8. Hence, 50% are the

   responses to the survey question twitching were below 8 and 50%

   of the responses were above 8. This is illustrated in Figure 2.


                iii
   Mode – 10

                                     14
                     BFS Statistical Analysis

The mode is the most common response or the response with the

highest occurrence or frequency. The most common answer for

twitching was 10 (the symptom happens 24/7). This is also

illustrated in Figure 2.


                                      iv
Standard Deviation—SD – 2.52


The standard deviation is a measure of the variability of a set of

responses around their mean. If responses cluster tightly around

the mean score, the standard deviation is smaller than it would be

with a more diverse group of responses from the mean. Any results

outside of the mean plus or minus three standard deviations is
                                                                     v
considered an outlier and discarded from the analysis. Figure 1

shows a common bell curve or what is sometimes referred to as a

normal distribution curve, probability density function, or Gaussian

distribution (µ is the mean and σ is the standard deviation). Figure

2 is the bell curve for the variable twitching. For twitching, the

standard deviation is 2.52 (3 standard deviations is equal to 7.6).

Hence, the mean plus 3 standard deviations is equal to 15.2 and

the mean minus 3 standard deviations is equal to 0. Obviously,

100% of the data responses for the twitching question lie within

this range since all answers had to be between 1 and 10 (no

outliers).


                                 15
                      BFS Statistical Analysis

Sample Size – n - 125


The sample size is equivalent to the number of people that

participated in the survey – 125. Remember, the sample size per

statistical test may be less than 125 because outliers were omitted

from the calculations. The exact sample size per variable is shown

on the “Data Summary” tab.

                               vi
Standard Error – SE - .230


Standard error is the standard deviation of the values of a given

function of the data (parameter), over all possible samples of the

same size. This is usually defined by the standard deviation (SD)

divided by the square root of the sample size (n). The smaller the

standard error the more tightly clustered the data results are

around the mean. And conversely, a high standard error means the

data distribution is widely dispersed around mean. One would

expect to find a large portion of the population (answers to the

twitching question) be between the mean plus and minus 3 times

the standard error.


                  vii
Variance – 6.37


The (population) variance of a random variable is a non-negative

number which gives an idea of how widely spread the values of the


                                    16
                    BFS Statistical Analysis

random variable are likely to be; the larger the variance, the more

scattered the observations are on average. In other words, variance

is a measure of the 'spread' of a distribution about its average

(mean) value. The variance for twitching is fairly dispersed because

responses covered the entire range of possibilities (1 through 10).

This too can be observed by reviewing Figure 2.


Percentile Rank – 1 at 0%, 5.875 at 25%, 8 at 50%, 10 at
                          viii
75%, and 10 at 100%

A percentile rank is typically defined as the proportion of scores in a

distribution that a specific score is greater than or equal to. For

percentile rank at 25%, this statistic equals the response where the

first 25% (frequency of occurrences) of the sample size population

resides. In the case of twitching 25% of the people answered 5.875

or lower. Obviously, the inverse is also true, that 75% of the people

answered higher than 5.875 for the variable twitching. Also, for

twitching, the percentile rank at 0% is 1, at 50% it is 8, at 75% it is

10, and at 100% it is also 10. This concept can be visualized in

Figure 2.



                                           ix
Inter – Quartile Range – IQR – 4.125




                                 17
                    BFS Statistical Analysis

The inter-quartile range is a measure of the spread of dispersion

within a data set. It is calculated by taking the difference between

the upper and the lower quartiles. IQR is generally defined as the

middle 50% of the data equal to percentile rank at 75% minus

percentile rank at 25%. In the case of the variable twitching

percentile rank at 75% = 10 and percentile rank at 25% = 5.75.

Hence, IQR equals 10 minus 5.875, which equals 4.25.


            x
Range – 9


The range of a sample (or a data set) is a measure of the spread or

the dispersion of the observations. It is the difference between the

largest and the smallest observed value of some quantitative

characteristic and is very easy to calculate. A great deal of

information is ignored when computing the range since only the

largest and the smallest data values are considered; the remaining

data are ignored. The range value of a data set is greatly influenced

by the presence of just one unusually large or small value in the

sample (outlier). In the twitching example the range of responses

were between 1 and 10. Hence, the range is equal to 10 minus 1,

which of course equals 9. Once again, this can be seen by viewing

the histogram for twitching in Figure 2.


                                             xi
Coefficient of Variation – CV – 33.03%

                                 18
                    BFS Statistical Analysis

The coefficient of variation (CV) measures the spread of a set of

data as a proportion of its mean. It is often expressed as a

percentage. It is the ratio of the sample standard deviation to the

sample mean. The smaller the coefficient of variance percentage

the more tightly clustered the result distribution is around the

mean. Conversely, the more dispersed a distribution is around the

mean equates to a larger coefficient of variance percentage. For

twitching the coefficient of variance result was 33.03%.


                    xii
Skewness - -.773


Qualitatively, a negative skew indicates that the tail on the left side

of the probability density function (Bell Curve) is longer than the

right side and the bulk of the values (possibly including the median)

lie to the right of the mean. A positive skew indicates that the tail

on the right side is longer than the left side and the bulk of the

values lie to the left of the mean. A zero value indicates that the

values are relatively evenly distributed on both sides of the mean,

typically but not necessarily implying a symmetric distribution. The

larger the absolute value of the skewness magnitude, the more

skewed the data is to the right or left (depending on the polarity) in

the bell curve. The twitching variable is skewed to the right as




                                 19
                                   BFS Statistical Analysis

   shown by Figure 2. The example in Figure 1 has no skewness

   because the bell curve is completely symmetrical.

   ______________________________________________________

   ______________________________________________________

   _____________________




Figure 1: Probability Density Function

                                                         Histogram
                  50
                                                                               Normal Fit
                  45                                                           (Mean=7.63,…
                  40
                  35
      Frequency




                  30
                  25
                  20
                  15
                  10
                  5
                  0
                       1   2   3   4   5       6     7      8    9   10   11
                                           Twitching



Figure 2: Probability Density Function (Bell Curve) for Twitching



                                                    20
                       BFS Statistical Analysis


                 n       119    (cases excluded: 5 due to missing values)

             Mean        7.63                            Median              8.00
            95% CI       7.17   to 8.09                95.7% CI              8.00   to 9.00
               SE       0.232
                                                          Range               9.0
          Variance       6.40                               IQR              4.42
               SD        2.53
           95% CI        2.24   to 2.90               Percentile
                                                            0th              1.00   (minimum)
                CV     33.2%                               25th              5.58   (1st quartile)
                                                           50th              8.00   (median)
         Skewness       -0.76                              75th             10.00   (3rd quartile)
           Kurtosis     -0.50                             100th             10.00   (maximum)


     Shapiro-Wilk W      0.85
                  p   <0.0001



Figure 3: Data Analysis for Twitching Variable


What can the data on the “Data Summary” tab tell us? It is very useful

to compare data for one parameter versus another parameter within

the same group or category of variables (ie symptoms, causes, body

parts, and remedies – these groups are color coded on the

spreadsheet). For symptom variables we can deduce the most

predictable parameter is Itching because it has the lowest standard

deviation, variance, and standard error. On the other hand, the

parameter Vibration/Buzzing Sensation has the lowest predictability

because it has the highest standard deviation, variance, and standard

error. This simply means that the responses to Itching are more tightly

distributed around the mean than the results to other parameters,

especially Vibration/Buzzing Sensation, which had sparsely distributed

                                          21
                       BFS Statistical Analysis

results around the mean. For Body Part affected variables the Chest

parameter was the most predictable while the Feet parameter was the

least predictable. For Remedy variables the Massage parameter (I did

not consider potassium channel drugs because very few people tried

them) was the most predictable while the Benzodiazepine variable was

the least predictable. This simply means that while benzodiazepine

drugs helped many people alleviate their symptoms, but at the same

time benzodiazepine drugs did very little to help other people alleviate

BFS symptoms creating a large variance in the distribution around the

mean result. This does not make benzodiazepine drugs a bad choice to

treat BFS. In fact, the opposite may be true since other remedies had

lower variances only because they did not work as well (in other

words, most people answered lowered numbered results for other

remedy variables such as sleeping pills, muscle relaxants,

homeopathic treatments, supplements, and so forth. Hence, the

results are clustered more tightly around a lower mean value).

Therefore, benzodiazepine drugs had more success than any other

treatment types because it had a higher mean, but the results were

mixed. In this study, parameters with lower means tended to have

lower variances, standard deviations, and standard errors. The data on

the “Data Summary” tab was calculated from the data on “BFS” tab




                                   22
                        BFS Statistical Analysis

(for all data except remedy variables) and the “BFS No Zero” tab (for

remedy variables).



Correlation Data:


The correlation results for all 125 responses can be found at:

http://patrickbohan.home.bresnan.net/BFS.htm. Click on the link

“Survey Data” and look at the excel file tab titled “Correlation Results”

to find statistical correlation data between variables (t-statistic data).
xiii




A linear regression model can have two purposes: one to predict future

results and or two, to find correlation. Most of the models generated
                                                   xiv
from the BFS survey have very low adjusted R²            values (the results

are not linear) and are therefore, not very good models to predict

future outcomes. On the other hand, linear regression models can give

us an idea of which parameters have strong or even weak correlation,

and that can be useful information. This is why the t-statistic result for

each model simulation is important because the t-statistic is a

measure of correlation.



Each question in the survey: your age, your sex, how bad you get pins

and needles, how well yoga works for you, etc is a variable or



                                     23
                       BFS Statistical Analysis

parameter (I use the two words interchangeably). When modeling

variables using a linear regression model, there are two sets of

variables - x and y. In the data result array (on the “Correlation

Results” tab) the horizontal axis is for y variables and the vertical axis

is for x variables (this is reversed from conventional algebra, but it

was easier for me to get the data into the table using this reversed

format). Only one variable is allowed for y in a linear regression

analysis, but multiple variables can be used for x (as long as there are

more equations than unknowns). For this study, I have grouped the x

variables into seven classifications – General (G), Causes / Triggers

(CA), Stressers (ST – those variables that can make BFS symptoms

worse), Symptoms (S), Body Parts affected (B), Remedies (RE), and

Various (V). I have used different color fonts to distinguish between

these groups on the “Correlation Results” tab for convenience. For

instance, the General (G) classification of variables consists of 7

parameters: age, sex, region, number of years with symptoms, years

diagnosed, EMG, and MRI.




The t-statistic is a good measure of correlation between corresponding

x and y variables. The higher the absolute value of the t-statistic

result, the better the correlation (lower standard error). A t-statistic

value of greater than 2 means very strong correlation; a t-statistic

                                    24
                        BFS Statistical Analysis

value between 1.75 and 2 means moderate correlation; and a t-

statistic value below 0.5 means the correlation is very weak.



The sign or polarity (+ or -) of the t-statistic result is also important. A

positive value means the x variable will tend to increase the value of

the y variable, whereas a negative value means the x variable will tend

to decrease the value of the y variable. For instance, a strong

correlation between twitching (y variable) and muscle relaxants (x

variable) can do one of two things: make the symptom better or worse

(- or + respectively). In this case, a positive value can increase the

twitching response (making it worse) whereas, a negative response

(would lower the twitching value – remember, twitching was rated on

a scale of 1 to 10 with 10 being the worse) would be beneficial and

something for people to try (as long as the magnitude of the t-statistic

showed strong correlation).



In essence, the “Correlation Results” tab is a matrix of t-statistic

results that is 57 long by 57 wide. T-statistic data was not obtained for

x variables within the same classification. For instance, Age as a y

variable was not modeled against other General (G) parameters such

as sex, region, years with symptoms etc. These results are designated

as “na” within the t-statistic matrix. Also, data in the matrix signified


                                     25
                        BFS Statistical Analysis

with ND (No Data) indicates the data was not linear dependent so no

results were computed. T-statistic results with strong correlation and a

positive polarity are in a bold green font. T-statistic results with strong

correlation and an negative polarity are signified with a bold red font.

T-statistic results with moderate correlation and a positive polarity are

in a bold blue font. And finally, t-statistic results with moderate

correlation and a negative polarity are signified with a bold orange

font.



One final note, I used the data on the “BFS” tab to model all results

except for Remedies (RE). When Remedy parameters were the y

variable I used the excel file tab “BFS No Zero” data to model the

results. After all, it does not make much sense to find correlation to

remedies that people have not tried (a “0” response means people did

not try the remedy). Hence, the data within the “BFS No Zero” tab is

the same as the data on the “BFS” tab except “0” responses to

Remedy questions were omitted from the data. But, it is important to

keep in mind, the model results of RE parameters using the “BFS No

Zero” tab will result in fewer data points (smaller sample size, n) in

the model. For this reason, the results from these models may prove

to be less conclusive because the data size is in some cases

significantly smaller. Hence, when evaluating the data models for RE


                                    26
                       BFS Statistical Analysis

correlation pay close attention to the sample size. When Remedies

(RE) are grouped together as the x variables, I use the data on the

“BFS” tab to run the models. Only a few people have tried all potential

remedies, hence the sample size would only be a single digit number if

the “BFS No Zero” tab data was used to model RE results as the x

variable.



Let’s examine the results of one y parameter, twitching. Six twitching

models were run using twitching as the y variable and G, CA, ST, B,

RE, and V classification of parameters as x variables respectively

(Figures 4 through Figure 9 respectively). The results listed below in

Table III provide t-statistic data for each parameter versus twitching.

Table III also contains a summary from the “Correlation” tab results to

include those parameters with the best correlation versus the listed

variable when it is modeled as the y variable:



Variable; t-statistic v. twitching; list of parameters with the best

correlation to the listed variable Green Font, (Red Font)




General Group (G)

Age; 1.17; cramps, acupuncture, (stress)

Sex; 0.98; (sensitivity to temperature), (yoga)


                                    27
                               BFS Statistical Analysis

Region; -0.23; stress, prescription drugs, history

Years Diagnosed (YD); -0.34; massage, altitude

Years with Symptoms (YBFS); 0.78; benzodiazepine, remedies,

(muscle pain)

EMG; 1.37; exercise, back, arms, anti-convulsants, muscle relaxants,

remedies, (stress1), (hip), (yoga)

MRI; -2.33; (chemicals), (twitching), remedies



                          n            120    (cases excluded: 5 due to missing values)


                          2
                         R            0.07
                           2
                Adjusted R            0.01
                         SE           2.51




                       Term     Coefficient            95% CI                      SE          t statistic    DF      p
                   Intercept          6.369        4.382 to 8.356                     1.0028           6.35    112   <0
                        Age        0.02713     -0.01878 to 0.07303                 0.023168            1.17    112    0
                        Sex           0.493       -0.499 to 1.485                     0.5008           0.98    112    0
                     Region       -0.04869     -0.46746 to 0.37009                 0.211356           -0.23    112    0
      Years Diagnosed (YD)        -0.02415     -0.16649 to 0.11820                 0.071840           -0.34    112    0
   Years with BFS Symptoms
                     (YBFS)        0.05059     -0.07735    to 0.17853              0.064572           0.78     112    0
                       EMG            0.751       -0.335   to 1.837                   0.5481          1.37     112    0
                        MRI          -1.196       -2.215   to -0.178                  0.5142         -2.33     112    0



Figure 4: Linear Regression Model: Twitching V. General (G)


There is strong negative correlation between twitching and a MRI. This

suggests people are more apt to get an EMG than an MRI due to

twitching symptoms.


                                               28
                                  BFS Statistical Analysis


Cause / Trigger Group (CA)

Flu Shot; -0.52; potassium channel

Chemicals; 2; (MRI), twitching, (hip), missing

Prescription Drugs (PD); 1.21; (region), vibration, (hands), anti-

depressants, missing

Spine or Neck Injury (SNI); 0.34; no correlation

Sickness; -1.34; (years with symptoms), sickness1, (headaches),

hands, diet

Exercise; 0.31; exercise1

Stress / Anxiety (SA); -1.44; (EMG), (exercise1), stress1, itching,

head, benzodiazepine, (time)

History; 0.19; (region), years with symptoms

Other; 0.83; (abdomen), anti-convulsants, (potassium channel),

homeopathic, remedies

                       n           120    (cases excluded: 5 due to missing values)


                       2
                    R             0.09
                      2
           Adjusted R             0.02
                    SE            2.50




                    Term    Coefficient          95% CI                        SE         t statistic    DF      p
                Intercept         7.997      6.923 to 9.072                      0.5422          14.75    110   <0.0001
           Flu Shot (FS)       -0.4126     -1.9880 to 1.1628                    0.79496          -0.52    110    0.6048
              Chemicals           2.218      0.019 to 4.417                      1.1095           2.00    110    0.0481
Prescription Drugs (PD)         0.7057     -0.4523 to 1.8636                    0.58430           1.21    110    0.2297
   Spine or Neck Injury
                    (SNI)       0.2437     -1.1709     to 1.6582                0.71379          0.34     110    0.7335
               Sickness        -0.6925     -1.7146     to 0.3296                0.51575         -1.34     110    0.1822

                                                     29
                               BFS Statistical Analysis
             Exercise         0.1821     -0.9658    to 1.3301                 0.57925          0.31     110    0.7538
 Stress / Anxiety (SA)       -0.7645     -1.8178    to 0.2888                 0.53151         -1.44     110    0.1532
               History        0.1175     -1.0958    to 1.3308                 0.61224          0.19     110    0.8482
                 Other        0.5219     -0.7309    to 1.7748                 0.63219          0.83     110    0.4108



Figure 5: Linear Regression Model: Twitching V. Causes (CA)


There is strong positive correlation between twitching and exposure to

chemicals as a trigger for BFS. In other words, if exposure to

chemicals were the cause or trigger of the BFS ailment, expect

twitching to be a primary symptom.



Stressers Group (ST)

Stress Anxiety1 (SA1); -0.47; (EMG), prescription drugs, history,

stress, head, (anti-convulsants), (potassium channel), homeopathic,

benzodiazepine

Exercise1; 0.58; exercise, sensitivity to temperatures, (acupuncture)

Sickness1; 1.07; vibration, sleeping pills, muscle relaxants,

(acupuncture), (benzodiazepine), missing

                    n           120    (cases excluded: 5 due to missing values)


                   2
                 R              0.02
                   2
        Adjusted R             -0.01
                 SE             2.54




                Term     Coefficient           95% CI                       SE          t statistic    DF      p
            Intercept          7.325       5.965 to 8.684                      0.6864          10.67    116   <0.0001
           Sickness1        0.08613     -0.07269 to 0.24495                 0.080187            1.07    116    0.2850
           Exercise1        0.04567     -0.10983 to 0.20116                 0.078507            0.58    116    0.5619

                                                   30
                            BFS Statistical Analysis
   Stress / Anxiety 1
               (SA1)    -0.03896   -0.20454    to 0.12662   0.083602   -0.47   116   0.6421



Figure 6: Linear Regression Model: Twitching V. Stressers (ST)


There is no moderate or strong correlation between twitching and

stressers.



Symptoms Group (S)

Twitching – na; (MRI), chemicals, lower leg, arms, time, day

Pins and Needles (PN) – na; time, feet, sickness1,

Cramps – na; age, exercise, muscle relaxants, remedies, time

Muscle Fatigue and Weakness (MFW) – na; back, (yoga), time

Headaches – na; (exercise), (chest), head, (yoga)

Itching – na; lower leg, head

Numbness – na; anti-depressants

Muscle Stiffness (MS) – na; sickness, head

Vibration / Buzzing Sensation (VBS) – na; prescription drugs,

sickness1, abdomen

Muscle Pain / Soreness (MPS) – na; EMG, arms, time

Sensitivity to Temperatures (STT) – na; exercise1, (day)



Body Part Group (B)




                                              31
                                 BFS Statistical Analysis

Feet; 1.47; (years diagnosed), twitching, pins and needles, sleeping

pills

Lower Leg (LL); 5.68; (years diagnosed), years with symptoms,

twitching, cramps, diet, time

Upper Leg (UL); -0.58; exercise1, muscle fatigue, (potassium

channel), time

Hip / Buttock (HBR); 0.78; prescription drugs

Back; -0.53; muscle fatigue, altitude

Abdomen; 0.23; (region), (age)

Chest; -0.47; (potassium channel), muscle relaxants

Neck / Head (NH); -0.63; headaches, (day)

Hands; -1.1; (age), muscle relaxants

Arms / Shoulders (AS); 2.12; (potassium channel), (sleeping pills),

EMG

                        n           113    (cases excluded: 12 due to missing values)


                      2
                    R              0.38
                        2
           Adjusted R              0.32
                    SE             2.10




                   Term     Coefficient            95% CI                       SE         t statistic    DF      p
               Intercept            3.07        1.33 to 4.81                       0.877           3.50    102    0.0007
                    Feet        0.1225       -0.0429 to 0.2879                   0.08340           1.47    102    0.1449
         Lower Leg (LL)         0.4831        0.3144 to 0.6517                   0.08504           5.68    102   <0.0001
         Upper Leg (UL)       -0.06082      -0.26952 to 0.14788                 0.105217          -0.58    102    0.5645
   Hip / Buttock Region
                  (HBR)        0.09856      -0.15053     to 0.34765             0.125580          0.78     102    0.4344
                   Back       -0.06282      -0.29603     to 0.17039             0.117576         -0.53     102    0.5943
              Abdomen           0.0423       -0.3158     to 0.4004               0.18056          0.23     102    0.8153


                                                    32
                           BFS Statistical Analysis
                Chest     -0.0869    -0.4501    to 0.2763     0.18311   -0.47   102   0.6361
     Neck / Head (NH)    -0.06607   -0.27538    to 0.14323   0.105522   -0.63   102   0.5326
               Hands      -0.1052    -0.2954    to 0.0850     0.09590   -1.10   102   0.2753
  Arms / Shoulder (AS)     0.2446     0.0160    to 0.4731     0.11524    2.12   102   0.0362



Figure 7: Linear Regression Model: Twitching V. Body Part (B)


There is super strong positive correlation between twitching and the

lower leg as well as the arms and shoulders. This should not come as

much of surprise since the lower leg and the arms are two of the most

affected body parts from BFS symptoms. The bottom line is that BFS

sufferers will tend to have strong twitching symptoms in their lower

legs and arms.



Remedies Group (RE)

Anti-Convulsants (AC); 0; EMG, sickness1, muscle pain, (chest)

Anti-Depressants (AD); -1.11, numbness

Potassium Channel Drugs (PCD); -0.95; (age), (head), (upper leg),

(stress1), (exercise1), exercise

Sleeping Pills (SP); 0.18; region, hip, (remedies)

Muscle Relaxants (MR); -0.81; (chemicals), sickness, (muscle fatigue)

Homeopathic Treatments (HT); 0.99; numbness

Supplements; -0.83; exercise

Diet; 1.2; sickness, vibration

Acupuncture; -0.48; (sex); (time)


                                           33
                                  BFS Statistical Analysis

Massage; 0.1; no correlation

Yoga; 0.13; (sex); (EMG)

Benzodiazepine Drugs (BD); 0.39; age, exercise, stress1, (headaches)

                              n          116    (cases excluded: 9 due to missing values)


                            2
                          R              0.06
                              2
                 Adjusted R             -0.05
                          SE             2.54




                         Term     Coefficient            95% CI                       SE        t statistic    DF      p
                     Intercept          7.831        6.951 to 8.712                    0.4439          17.64    103   <0.0
                                                         -
       Anti-Convulsants (AC)
                                   0.0006226    0.2734014 to 0.2746467            0.13816823           0.00     103    0.9
      Anti-Depressants (AD)           -0.1761      -0.4897 to 0.1376                 0.15814          -1.11     103    0.2
   Potassium Channel Drugs
                       (PCD)         -0.6713      -2.0695     to 0.7270               0.70503         -0.95     103    0.3
          Sleeping Pills (SP)        0.02569     -0.25187     to 0.30325             0.139951          0.18     103    0.8
     Muscle Relaxants (MR)           -0.1367      -0.4700     to 0.1967               0.16808         -0.81     103    0.4
Homeopathic Treatments (HT)           0.2113      -0.2121     to 0.6347               0.21347          0.99     103    0.3
               Supplements           -0.1376      -0.4673     to 0.1921               0.16623         -0.83     103    0.4
                          Diet        0.2273      -0.1493     to 0.6039               0.18989          1.20     103    0.2
               Acupuncture           -0.1221      -0.6235     to 0.3794               0.25282         -0.48     103    0.6
                   Massage           0.01742     -0.31329     to 0.34814             0.166753          0.10     103    0.9
                        Yoga         0.02319     -0.33952     to 0.38589             0.182884          0.13     103    0.8
 Benzodiazepine Drugs (BD)           0.04001     -0.16197     to 0.24199             0.101842          0.39     103    0.6



Figure 8: Linear Regression Model: Twitching V. Remedies (RE)

There is no moderate or strong correlation to suggest any remedies

work very well to alleviate the twitching symptom.



Various Group (V)

Remedies; -0.4; EMG, sickness1, cramps, back, massage

Day; 2.27; twitching, (sensitivity to temperatures), (head)



                                                   34
                              BFS Statistical Analysis

Time; 4.86; (stress), exercise1, twitching, cramps, lower leg

Missing; 1.91; prescription drugs, history, sickness1, back, anti-

convulsants

Altitude; 0.05; years diagnosed

                    n          120    (cases excluded: 5 due to missing values)


                  2
                R             0.22
                    2
       Adjusted R             0.18
                SE            2.28




              Term      Coefficient            95% CI                      SE          t statistic    DF      p
          Intercept           3.753        2.169 to 5.337                     0.7996           4.69    114   <0.0001
         Remedies          -0.2534      -1.5071 to 1.0004                   0.63289           -0.40    114    0.6897
               Time         0.5327       0.3154 to 0.7500                   0.10972            4.86    114   <0.0001
                 Day        0.5712       0.0724 to 1.0701                   0.25182            2.27    114    0.0252
           Missing            1.379       -0.053 to 2.811                     0.7228           1.91    114    0.0590
            Altitude       0.04856     -1.77122 to 1.86835                 0.918623            0.05    114    0.9579



Figure 9: Linear Regression Model: Twitching V. Various (V)



There is super strong positive correlation between twitching and how it

affects us during the day and over time. Once again, this should come

as no surprise since BFS sufferers feel twitching 76.4% of the time.

Oddly, there is also moderate correlation between missing remedies

from the survey and making twitching symptoms worse. I cannot

explain this result.




                                                 35
                       BFS Statistical Analysis


Data Naming Convention:


Since it is impossible to input all the statistical data and graphs (Figure

2 and Figure 3) as well as all the modeling data (Figures 4 through

Figure 9) into this paper for all parameters (there are nearly 400 data

summaries and models), the information can be obtained from my BFS

website. I will keep the survey open and update the information

periodically. The data responses for each parameter (similar to Figure

2 and Figure 3) and linear regression models (similar to Figures 4

through 9) can be found at:

http://patrickbohan.home.bresnan.net/BFS.htm and click on the link

“BFS Data Summary” and “BFS Correlation Summary” respectively.

Each tab on these excel files has a unique name and represents the

statistical data summary of one variable or one model simulation. The

tab naming convention used on the “BFS Data Summary” link is the

parameter name and the extension DA (short for Data). Hence, the tab

name Age-DA will contain a statistical data summary of the Age

parameter. In some cases, the name is abbreviated such as YD-DA is

short for Years Diagnosed or YBFS-DA is short for years with BFS

Symptoms. For Remedy (RE) parameters I used the “BFS No Zero” tab

to compute the data summary results (to eliminate “0” responses

where people never tried the remedy). The naming convention on the


                                    36
                       BFS Statistical Analysis

excel file tabs is, for example, Diet-NZDA (NZDA stands for No Zero

Data). If the –DA and –NZDA results do not equal the results on the

“Data Summary” tab of the “Survey Data” excel file, it is because the

information on the “Survey Data” is updated immediately when the

data is downloaded. The –DA and –NZDA data pages are not updated

as often. The tab naming convention for the “BFS Correlation

Summary” link is: y parameter–x parameter group. For this study, the

x variables are grouped into seven classifications – General (G),

Causes / Triggers (CA), Stressers (ST – those variables that can make

BFS symptoms worse), Symptoms (S), Body (B), Remedies (RE), and

Various (V). For instance, the General (G) variables consist of 7

parameters: age, sex, region, number of years with symptoms (YBFS),

years diagnosed (YD), EMG, and MRI. Hence, the tab names for Age

models (when Age is the y variable) are Age-CA, Age-ST, Age-S, Age-

B, Age-RE, and Age-V. In some cases I abbreviated the y variable

names such as YD-CA, YD-ST, YD-S, YD-B, YD-RE, and YD-V (YD is for

Years Diagnosed).


Data Results:


Do the data results make sense? This is a difficult question to answer,

but it is one that scientists, engineers, and mathematicians must try to

address. An eyeball test of some results entered by participants does



                                   37
                       BFS Statistical Analysis

not make sense to me. For instance, some people scored their

symptoms and body parts affected by BFS very high (well above

average), but at the same time they claimed that certain remedies

were very helpful. People that scored remedies usefulness for

example, an 8, on a scale of 10, I would have expected that their

symptom average to be no higher than 2. In other words, I would

expect these two categories to be inversely proportional, but that was

not necessarily the case for all respondents. That is the problem with a

subjective questionnaire; each person has its own interpretation of the

questions. After all, there are no right or wrong answers, but some of

these responses were omitted as outliers.



There were a few models where one would expect to find strong

correlation between certain x and y parameters. For instance, people

who felt the cause / trigger of their BFS symptoms were stress,

exercise, or sickness, I would therefore naturally hypothesize stress,

exercise, and sickness would make their symptoms worse once they

were stricken with the BFS disorder. So, let’s examine the correlation

models between Causes (CA) and Stressers (ST) to see if this

hypothesis is true. To do so, there are six models we need to evaluate:

Exercise-ST, SA-ST, Sickness-ST, Exercise1-CA, SA1-CA, and

Sickness1-CA (SA is short for Stress / Anxiety). The ST model results


                                   38
                       BFS Statistical Analysis

are what one would suspect. Exercise-ST showed strong correlation

between people who felt their illness was caused / triggered by

exercise and therefore, exercise makes their BFS symptoms worse.

Conversely, SA-ST and Sickness-ST showed strong correlation

between people who felt their BFS illness was caused / triggered by

stress and or a sickness did indeed show that stress and or a sickness

made their symptoms worse, respectively. The SA1-CA and Exercise1-

CA models were also predictable. They showed very strong correlation

between stress making their symptoms worse and stress and exercise

causing / triggering the onset of their BFS illness respectively.

However, the Sickness1-CA model only showed moderate correlation

between sickness making their symptoms worse and sickness causing

/ triggering the onset of their illness (but the sickness variable had the

strongest correlation of all cause (CA) variables. It is also important to

note that the model results for people where stress makes their

symptoms worse also indicates that exercise could help relieve their

symptoms.



Here are a few other correlation model results that make some sense:

only women have primarily used yoga as a remedy; cramping tends to

get worse the older we become; older people primarily have tried




                                    39
                       BFS Statistical Analysis

acupuncture; pins and needles primarily occur in our feet; and people

with strong BFS symptoms in their head tend to have headaches.



The analysis of the above 6 models does, for the most part, illustrate

that data responses for these questions does in fact make sense.

Other model categories would be much harder to deduce if the data

makes sense. However, if the results for some questions make sense

then one can be fairly confident that other model category results are

most likely, fairly accurate. At least we can hope they are.



Future Studies:

It is possible to expand this study in the future. For instance, it was

learned that supplements were a good treatment for BFS sufferers

whose symptoms are caused by exercise. However, we do not know

what supplements that people used so this result can be vague and

misleading. However, a future study can pin point what supplements

people used (for example, magnesium, potassium, vitamin D, and

quinine are all common supplements tried by BFS sufferers).



Acknowledgments:


I would like to thank my fellow BFS suffers for taking part in this

survey and study. Without their cooperation we would have failed to

                                    40
                       BFS Statistical Analysis

obtain a decent sample size to conduct this study and subsequently

bring forth pertinent statistical information about the BFS ailment.



Conclusions:


The best way to characterize a person with BFS is as follows (those

parameters with an average above 65% on Table I): A North American

male about 39 years old who has had symptoms for about 3.5 years

and has been diagnosed with BFS for about 2.25 years. They had a

MRI to prove they do not have MS. This person believes that stress

caused or triggered the symptoms and stress will most likely make

symptoms worse. The primary symptom is twitching in the lower leg

and most remedies and drugs do very little to alleviate the symptoms.



There was a pattern on the “Data Summary” tab for parameters whose

responses were between 1 and 10. Generally speaking, parameters

with high averages (7 or higher) or parameters with low averages (3

or lower) tended to have lower standard deviations, variances, and

standard errors than parameters with medium averages (4 through 6).

This tells us that parameters with medium averages had mixed results

(some people answered 10 while others answered 1). This means

these parameters are less predictable and do not necessarily define

BFS symptoms. For instance, the results for Symptom (S) variables

                                   41
                       BFS Statistical Analysis

illustrates that Twitching (high average) and Itching (low average) had

lower variances (tighter distribution – more predictable) than medium

average symptoms such as Muscle Stiffness (MS), Muscle Pain and

Soreness (MPS), Vibration/Buzzing Sensation (VBS), and Muscle

Fatigue and Weakness (MFW). Hence, BFS can be better defined by

saying sufferers are most likely to exhibit muscle twitching symptoms,

but no itching symptoms, whereas MS, MPS, VBS, and MFW may be a

symptom in some sufferers but not in others – less predictable.



The summary of the strong correlation results in Table III can tell us a

lot about the BFS disorder. One thing that sticks out is that 5 of the 11

symptoms showed strong positive correlation to the variable time

indicating the symptoms are becoming worse over time. In fact, BFS

symptoms can get worse over time unless the disorder was triggered

by stress and made worse by stress. Stress is the one thing we can

control to minimize symptoms even though no remedy seems to work

to alleviate stress. In fact, of the nine potential causes only exercise

(benzodiazepines and supplements) and sickness (diet and muscle

relaxants) show some success with remedies (but remember the

amount of relief from these remedies is usually very small – low

statistical averages). We may also be able to draw some other

conclusions from Table III: prescription drugs causing or triggering


                                    42
                      BFS Statistical Analysis

BFS leads to the vibration / buzzing symptom; anti-depressants may

be the best remedy to treat the symptom numbness; the itching

symptom primarily occurs on the head; muscle fatigue primarily occurs

on the back; sensitivity to temperatures is primarily caused and

triggered by exercise; exercise may help alleviate headaches; and so

forth.



I am not a doctor, but I am an expert on BFS because I am inflicted

with the disorder. I am not a PhD, but I have worked extensively on

data analysis and modeling over my 22 year career as an engineer. I

would offer my services to collect and analyze data for any disorder

free of charge (contact me at pbohan1@gmail.com), as long as

someone will generate the questionnaire or survey. The goal of this

paper is to not only better define and understand BFS, but to give it

the exposure it deserves. At a minimum, if the information in this

study can provide some sense of comfort to the people inflicted with

BFS, then it accomplishes its goal. Remember, stress is a big trigger

and can inflate symptoms – and having BFS creates unneeded stress –

People with BFS are always asking themselves morbid questions: Do I

have MS? Do I have ALS? Am I going to die? If we can alleviate these

fears by showing others are going through the same situation, then we

accomplished one major goal in this writing.


                                   43
                                  BFS Statistical Analysis

i
   Boslaugh, Sarah, Watter, Andrew Paul, Statistics in a Nutshell, (O’Reilly Media Corp, 2008), page 56
ii
    Boslaugh, Watter, Statistics in a Nutshell, page 57
iii
    Boslaugh, Watter, Statistics in a Nutshell, page 58
iv
    Boslaugh, Watter, Statistics in a Nutshell, page 60
v
    Boslaugh, Watter, Statistics in a Nutshell, page 64
vi
    Boslaugh, Watter, Statistics in a Nutshell, page 174
vii
     Boslaugh, Watter, Statistics in a Nutshell, page 162
viii
     Boslaugh, Watter, Statistics in a Nutshell, page 368
ix
    Boslaugh, Watter, Statistics in a Nutshell, page 370
x
    Boslaugh, Watter, Statistics in a Nutshell, page 59
xi
    Boslaugh, Watter, Statistics in a Nutshell, page 62
xii
     Boslaugh, Watter, Statistics in a Nutshell, page 58
xiii
     Boslaugh, Watter, Statistics in a Nutshell, page 179
xiv
     Boslaugh, Watter, Statistics in a Nutshell, page 182




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