Biostatistics basics - descriptive statistics

Biostatistics Basics An introduction to an expansive and complex field

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Common statistical terms • Data – Measurements or observations of a variable

• Variable – A characteristic that is observed or manipulated – Can take on different values

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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 Evidence-based Chiropractic

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Statistical terms (cont.) • Parameters – Summary data from a population

• Statistics – Summary data from a sample

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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 Evidence-based Chiropractic

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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. Evidence-based Chiropractic

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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 Evidence-based Chiropractic

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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

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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 Evidence-based Chiropractic

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Hypothetical study data (partial from book) • Distribution provides a summary of: – Frequencies of each of the values • • • • • •

2–3 3–4 4–3 5–1 6–1 7–2

etc.

– Ranges of values • Lowest = 2 • Highest = 7 Evidence-based Chiropractic

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Case # 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Visits 7 2 2 3 4 3 5 3 4 6 2 3 7 4 © 2006

Frequency distribution table

•2 •3 •4 •5 •6 •7

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Frequency Percent 3 21.4 4 28.6 3 21.4 1 7.1 1 7.1 2 14.3

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Cumulative % 21.4 50.0 71.4 78.5 85.6 100.0

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Frequency distributions are often depicted by a histogram

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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. Evidence-based Chiropractic

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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

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Formula to calculate the mean X n

• Mean of a sample

X 

• Mean of a population

X  N

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

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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

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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

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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 Evidence-based Chiropractic

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Levels of measurement •

There are 4 levels of measurement – Nominal, ordinal, interval, and ratio

1. 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)

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Levels of measurement (cont.) 2. 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)

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Levels of measurement (cont.) • 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 Evidence-based Chiropractic

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Levels of measurement (cont.) 3. 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

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Levels of measurement (cont.) 4. 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

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Levels of measurement (cont.) • 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 Evidence-based Chiropractic

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

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Levels of measurement (cont.)

Measurement scale

Permissible mathematic operations

Best measure of central tendency

Nominal

Counting

Mode

Ordinal

Greater or less than operations

Median

Interval

Addition and subtraction

Symmetrical – Mean Skewed – Median

Ratio

Addition, subtraction, multiplication and division

Symmetrical – Mean Skewed – Median

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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) Evidence-based Chiropractic

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The shape of data (cont.)

Line depicting the shape of the data

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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 Evidence-based Chiropractic

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The normal distribution (cont.) Mean The highest point of the overlying normal curve is at the mean

As one moves away from the mean in either direction the height of the curve decreases, approaching, but never reaching zero

A normal distribution is symmetric about its mean

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The normal distribution (cont.) Mean = Median = Mode

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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 Evidence-based Chiropractic

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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

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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 Evidence-based Chiropractic

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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

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More properties of normal curves (cont.)

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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 Evidence-based Chiropractic

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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

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SD (cont.) Ages are spread out along an X axis

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

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Distances ages deviate above and below the mean

Etc.

Adding deviations always equals zero

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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

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Calculating S2 cont.

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

Symbol for SD of a sample  for a population

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Calculating SD with Excel

Enter values in a column

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SD with Excel (cont.)

Click Data Analysis on the Tools menu

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SD with Excel (cont.)

Select Descriptive Statistics and click OK

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SD with Excel (cont.)

Click Input Range icon

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SD with Excel (cont.)

Highlight all the values in the column

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SD with Excel (cont.)

Click OK

Check if labels are in the first row Check Summary Statistics

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SD with Excel (cont.)

SD is calculated precisely Plus several other DSs

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Wide spread results in higher SDs narrow spread in lower SDs

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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

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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  z  Evidence-based Chiropractic

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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 Evidence-based Chiropractic

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z-scores (cont.) Refer to a z-table to find proportion under the curve

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

z-scores (cont.)

Z

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.0

0.5000

0.5040

0.5080

0.5120

0.5160

0.5199

0.5239

0.5279

0.5319

0.1

0.5398

0.5438

0.5478

0.5517

0.5557

0.2

0.5793

0.5832

0.5871

0.5910

0.3

0.6179

0.6217

0.6255

0.6293

0.5596 0.5636 0.5675 Corresponds to the 0.5714 area 0.5753 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 under the curve in black 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

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

0.9332 0.9332 0.9345

0.9357

0.9370

0.9382 55

0.9394

0.9406

0.9418

0.9429

0.9441

1.5

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0.09 0.5359

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