Biostatistics basics-biostatistics4734

Biostatistics Basics

An introduction to an expansive
and complex field

1
© 2006

Common statistical terms

• Data
– Measurements or observations of a variable
• Variable
– A characteristic that is observed or
manipulated
– Can take on different values

E vid e nce -b as e d C h irop ractic 2 © 2006

Statistical terms (cont.)

• Independent variables
– Precede dependent variables in time
– Are often manipulated by the researcher
– The treatment or intervention that is used in a
study
• Dependent variables
– What is measured as an outcome in a study
– Values depend on the independent variable

Statistical terms (cont.)

• Parameters
– Summary data from a population
• Statistics
– Summary data from a sample


Populations

• A population is the group from which a
sample is drawn
– e.g., headache patients in a chiropractic
office; automobile crash victims in an
emergency room
• In research, it is not practical to include all
members of a population
• Thus, a sample (a subset of a population)
is taken

Random samples

• Subjects are selected from a population so
that each individual has an equal chance
of being selected
• Random samples are representative of the
source population
• Non-random samples are not
representative
– May be biased regarding age, severity of the
condition, socioeconomic status etc.

Random samples (cont.)

• Random samples are rarely utilized in
health care research
• Instead, patients are randomly assigned to
treatment and control groups
– Each person has an equal chance of being
assigned to either of the groups
• Random assignment is also known as
randomization

Descriptive statistics (DSs)

• A way to summarize data from a sample
or a population
• DSs illustrate the shape, central tendency,
and variability of a set of data
– The shape of data has to do with the
frequencies of the values of observations


DSs (cont.)

– Central tendency describes the location of the
middle of the data
– Variability is the extent values are spread
above and below the middle values
• a.k.a., Dispersion
• DSs can be distinguished from inferential
statistics
– DSs are not capable of testing hypotheses


Hypothetical study data
(partial from book)
Case # Visits
• Distribution provides a summary of: 1 7
2 2
– Frequencies of each of the values 3 2
• 2–3 4 3
• 3–4 5 4
• 4–3 etc. 6 3
7 5
• 5–1 8 3
• 6–1 9 4
• 7–2 10 6
11 2
– Ranges of values 12 3
• Lowest = 2 13 7
• Highest = 7 14 4


Frequency distribution table

Frequency Percent Cumulative %
•2 3 21.4 21.4
•3 4 28.6 50.0
•4 3 21.4 71.4
•5 1 7.1 78.5
•6 1 7.1 85.6
•7 2 14.3 100.0


Frequency distributions are
often depicted by a histogram


Histograms (cont.)

• A histogram is a type of bar chart, but
there are no spaces between the bars
• Histograms are used to visually depict
frequency distributions of continuous data
• Bar charts are used to depict categorical
information
– e.g., Male–Female, Mild–Moderate–Severe,
etc.

Measures of central tendency

• Mean (a.k.a., average)
– The most commonly used DS
• To calculate the mean
– Add all values of a series of numbers and
then divided by the total number of elements


Formula to calculate the mean

• Mean of a sample ΣX
X =
n

• Mean of a population ΣX
µ=
N
• X (X bar) refers to the mean of a sample and μ refers to the
mean of a population
• m is a command that adds all of the X values
X
• n is the total number of values in the series of a sample and
N is the same for a population


Measures of central
tendency (cont.)

• Mode Mode

– The most frequently
occurring value in a
series
– The modal value is
the highest bar in a
histogram


Measures of central
tendency (cont.)

• Median
– The value that divides a series of values in
half when they are all listed in order
– When there are an odd number of values
• The median is the middle value
– When there are an even number of values
• Count from each end of the series toward the
middle and then average the 2 middle values


Measures of central
tendency (cont.)

• Each of the three methods of measuring
central tendency has certain advantages
and disadvantages
• Which method should be used?
– It depends on the type of data that is being
analyzed
– e.g., categorical, continuous, and the level of
measurement that is involved


Levels of measurement

• There are 4 levels of measurement
– Nominal, ordinal, interval, and ratio
2. Nominal
– Data are coded by a number, name, or letter
that is assigned to a category or group
– Examples
• Gender (e.g., male, female)
• Treatment preference (e.g., manipulation,
mobilization, massage)


Levels of measurement (cont.)

1. Ordinal
– Is similar to nominal because the
measurements involve categories
– However, the categories are ordered by rank
– Examples
• Pain level (e.g., mild, moderate, severe)
• Military rank (e.g., lieutenant, captain, major,
colonel, general)



• Ordinal values only describe order, not
quantity
– Thus, severe pain is not the same as 2 times
mild pain
• The only mathematical operations allowed
for nominal and ordinal data are counting
of categories
– e.g., 25 males and 30 females



1. Interval
– Measurements are ordered (like ordinal
data)
– Have equal intervals
– Does not have a true zero
– Examples
• The Fahrenheit scale, where 0° does not
correspond to an absence of heat (no true zero)
• In contrast to Kelvin, which does have a true zero



1. Ratio
– Measurements have equal intervals
– There is a true zero
– Ratio is the most advanced level of
measurement, which can handle most types
of mathematical operations



• Ratio examples
– Range of motion
• No movement corresponds to zero degrees
• The interval between 10 and 20 degrees is the
same as between 40 and 50 degrees
– Lifting capacity
• A person who is unable to lift scores zero
• A person who lifts 30 kg can lift twice as much as
one who lifts 15 kg



• NOIR is a mnemonic to help remember
the names and order of the levels of
measurement
– Nominal
Ordinal
Interval
Ratio



Permissible mathematic Best measure of
Measurement scale
operations central tendency

Nominal Counting Mode

Greater or less than
Ordinal Median
operations

Symmetrical – Mean
Interval Addition and subtraction
Skewed – Median

Addition, subtraction, Symmetrical – Mean
Ratio
multiplication and division Skewed – Median


The shape of data

• Histograms of frequency distributions have
shape
• Distributions are often symmetrical with
most scores falling in the middle and fewer
toward the extremes
• Most biological data are symmetrically
distributed and form a normal curve (a.k.a,
bell-shaped curve)

The shape of data (cont.)

Line depicting
the shape of
the data


The normal distribution

• The area under a normal curve has a
normal distribution (a.k.a., Gaussian
distribution)
• Properties of a normal distribution
– It is symmetric about its mean
– The highest point is at its mean
– The height of the curve decreases as one
moves away from the mean in either direction,
approaching, but never reaching zero

The normal distribution (cont.)

Mean
The highest point of
As one moves away from
the overlying
the mean in either direction
normal curve is at
the height of the curve
the mean
decreases, approaching,
but never reaching zero

A normal distribution is symmetric about its mean


The normal distribution (cont.)

Mean = Median = Mode


Skewed distributions

• The data are not distributed symmetrically
in skewed distributions
– Consequently, the mean, median, and mode
are not equal and are in different positions
– Scores are clustered at one end of the
distribution
– A small number of extreme values are located
in the limits of the opposite end


Skewed distributions (cont.)

• Skew is always toward the direction of the
longer tail
– Positive if skewed to the right
– Negative if to the left

The mean is shifted
the most


Skewed distributions (cont.)

• Because the mean is shifted so much, it is
not the best estimate of the average score
for skewed distributions
• The median is a better estimate of the
center of skewed distributions
– It will be the central point of any distribution
– 50% of the values are above and 50% below
the median

More properties
of normal curves
• About 68.3% of the area under a normal
curve is within one standard deviation
(SD) of the mean
• About 95.5% is within two SDs
• About 99.7% is within three SDs


More properties
of normal curves (cont.)


Standard deviation (SD)

• SD is a measure of the variability of a set
of data
• The mean represents the average of a
group of scores, with some of the scores
being above the mean and some below
– This range of scores is referred to as
variability or spread
• Variance (S2) is another measure of
spread

SD (cont.)

• In effect, SD is the average amount of
spread in a distribution of scores
• The next slide is a group of 10 patients
whose mean age is 40 years
– Some are older than 40 and some younger


SD (cont.)

Ages are spread
out along an X axis

The amount ages are
spread out is known as
dispersion or spread


Distances ages deviate above
and below the mean

Etc.

Adding deviations
always equals zero


Calculating S2

• To find the average, one would normally
total the scores above and below the
mean, add them together, and then divide
by the number of values
• However, the total always equals zero
– Values must first be squared, which cancels
the negative signs


Calculating S2 cont.

S2 is not in the
same units (age),
but SD is

Symbol for SD of a sample
σ for a population


Calculating SD with Excel

Enter values in a column


SD with Excel (cont.)

Click Data Analysis
on the Tools menu



Select Descriptive
Statistics and click OK



Click Input Range icon



Highlight all the
values in the column



Click OK

Check if labels are
in the first row

Check Summary
Statistics



SD is calculated precisely
Plus several other DSs


Wide spread results in higher SDs
narrow spread in lower SDs


Spread is important when
comparing 2 or more group means

It is more difficult to
see a clear distinction
between groups
in the upper example
because the spread is
wider, even though the
means are the same


z-scores

• The number of SDs that a specific score is
above or below the mean in a distribution
• Raw scores can be converted to z-scores
by subtracting the mean from the raw
score then dividing the difference by the
SD
X −µ
z=
σ

z-scores (cont.)

• Standardization
– The process of converting raw to z-scores
– The resulting distribution of z-scores will
always have a mean of zero, a SD of one, and
an area under the curve equal to one
• The proportion of scores that are higher or
lower than a specific z-score can be
determined by referring to a z-table

z-scores (cont.)

Refer to a z-table
to find proportion
under the curve


Partial z-table (to z = 1.5) showing proportions of the
z-scores (cont.)
area under a normal curve for different values of z.

Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517
Corresponds to the 0.5714 0.5753
0.5557 0.5596 0.5636 0.5675
area
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293
under the curve in black 0.6517
0.6331 0.6368 0.6406 0.6443 0.6480
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345
0.9332 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

Biostatistics basics-biostatistics4734

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