Statistics & Their Use

OBJECTIVES Understand the reason for and use of

statistics Review descriptive statistics

Measures of central tendency Measures of variability Measures of relationship

Inferential Statistics Parametric Non-parametric

What are statistics for?Painting a mathematical picture….or

simplifying large sets of data Identifying if relationships existEstablishing the probability of a cause

and effect relationshipETC…..

How many tests are there?There are well over 100 statistical

calculations and tests

However, many are rarely if ever used

Therefore, we will focus on those that are used most often…

Which tests should you know?

The New England Journal of Medicine from Vol 298 through 301 – (760 articles) were reviewed to determine what statistical tests are most important to learn to understand the scientific literature.

FINDINGS “A reader who is conversant with some

simple descriptive statistics (percentages, means, and standard deviations) has full statistical access to

58% of the articles.

Findings “Understanding t-test increases this

access to 67 percent. “Familiarity with each additional test

gradually increases the percentage of accessible articles.”

Descriptive StatisticsPurpose --- To simply describe things

the way they areNOT to establish a possible cause &

effect relationship

Inferential StatisticsPurpose --- In experimental research

uses a sample of the population – inferential statistics permits the researcher to generalize from the sample data to the entire population.

Aids the researcher in determining if cause and effect relationships exist.

Descriptive Statistics

Measures of Central Tendency

Mode --- most frequently occurring score

Median (Mdn) --- score physically in the middle of all scores

Mean (M or X) --- arithmetic mean--- i.e. sum of scores divided by the number of scores

Mean Is generally the preferred measure of

central tendency Is used frequently for other calculations

Means of 2 data sets

Measures of Variability (dispersion)

Measures of Variability If scores are similar….ergo they have low

variability (homogeneous) If scores are dissimilar…ergo they have high

variability (heterogeneous) Two sets of scores may have the exact same

mean but one set may have low variability and the other very high….therefore… measures of variability help DESCRIBE these differences

RangeSimplest measure of variabilityThe difference between the lowest and

highest scoreUsually reported in the literature as

“range” but on occasion as “R”The range is an unstable calculation

because it is only based upon 2 scores

Standard Deviation & VarianceThese measures are calculated based

upon ALL data scores and are therefore better represent the data set

They are used with other statistical calculations

Variance Related to the amount that an individual

score VARIES from the mean.

Standard Deviation Is the square root of the variance

Standard Deviation (aka S.D.)

CorrelationsMeasures of central tendency and

variability describe only ONE variableCORRELATIONS describe the

relationship between TWO variablesCorrelations can be positive, negative or

zeroCorrelations range from +1 through -1

Correlations+.95, +.87, High positive correlations+.19, +.22, Low positive correlations+.03, -.02, No relationship -.23, -.19, Low negative correlations -.94, -.88, High negative correlations

Spearman’s Rho & PearsonPearson’s product-moment correlation

is a parametric test (for continuous data….blood pressure, weight, etc.)

Spearman’s Rho is non-parametric ( for rank data…male/female, etc.)

Kappa value ---used often for determining degree of inter and intra-examiner agreement

Kappa Commonly used in chiropractic & medical literature

to convey the degree of interexaminer and/or intraexaminer reliability

Interexaminer– two or more examiners checking/evaluating or testing for the same finding

Intraexaminer– one examiner checking/evaluating or testing for the same finding on two different occasions

Kappa calculates the degree of agreement between the first and second check/evaluation or test

Summary Descriptive statistics do just that They describe the distribution of data toward

the center (mean, median, mode) and they describe the variability away from the center (range, variance, standard deviation)

They also determine if there is a relationship between 2 (or more) variables

Summary Of the 100+ statistical tests only a few are

frequently used You can intelligently read and understand

nearly 70% of the biomedical literature with an understanding of descriptive statistics and the t-test

Correlations are an important and often used type of descriptive statistics

Descriptive Statistics must be well understood in order to understand inferential statistics

Making Too Much from Research (even if it is well controlled research)

Inferential Statistics

Parametric versus Nonparametric

Assumptions---Parametric Parametric Inferential Statistics --- assumes

that the sample comes from a population that is NORMALLY DISTRUBUTED & that the variance is similar (homogeneous) between sample and population (or 2 populations)

The tests are very POWERFUL ---I.e. can recognize if there is a significant change based upon the experimental manipulation

Nonparametric Inferential Statistics --- no AssumptionsMakes no assumptions about the

distribution of the data (distribution free)So it does not assume that there is a

normal distribution of the data…..etc. Is less powerful…meaning that a

greater difference (or change) needs to be present in the data before a significant difference can be detected

Usage “Generally, it is agreed that unless there

is sufficient evidence to suggest that the population is extremely non-normal and that the variances are heterogeneous, parametric tests should be used because of their additional power”

Parametric Statistics

3 most commonly used tests

& some related concepts commonly expressed in the literature

Statistical & Medical Significance It is important to keep in mind that

statisticians use the word “significance” to represent the results of testing a hypothesis

In everyday language and in the clinical setting, a “significant” finding or treatment relates to how “important” it is from a clinical and not a mathematical perspective

4 Possibilities Medically & statistically significant Medically but not statistically significant Statistically but not medically significant Neither statistically or medically significant

Very large groups of subjects can reflect statistically significant differences between two groups …but they may not be medically significant from the perspective of cost, risks, policies etc.

Parametric Statistical Tests

t-Test aka Student t-Test

ANOVA

Analysis of covariance

t- Test Developed by Gosset under the pseudonym

Student 3 different versions of the t-tests that apply to

3 different research designs All three forms of the t-Test are based upon

the MEANS of two groups The larger the difference in the calculated t

scores, the greater the chance that the null hypothesis can be rejected

t-Test3 different general ways to use the

student t-test2 variations of each…dependent upon

the type of hypothesis the researcher uses

The hypothesis can either be directional or non-directional

Directional / Non-directional Hypothesis “Directional” means that the researcher

anticipates or expects a specific positive or negative impact from the treatment (or other independent variable)

“Non-directional” means that the researcher does not know what to expect. Perhaps the treatment will make the patient better or worse

Critical Value The researcher must establish the value at

which they will consider the results “significant”…this is referred to as the CRITICAL VALUE

There is some subjective and somewhat arbritrary decision to be made in this regard by the researcher

The customary critical values are either P<.05 or P<.01 but on occasion you will see P<.10

Directional Hypothesis/Non… & Critical ValueThese two decisions need to be made

before the study is started and will determine to some degree if your results will be significant

…

t-Test #1 Single Sample Compares the sample to the mean value of

the population Is not used often because the mean for the

population if usually not known E.g. Stanford Binet I.Q. test….has a mean of

100 and 1 S.D. of 16 (although not everyone in the U.S. has had the test, enough have been tested to accept the data as representative

t-Test #2 Correlated Groups Used when subjects serve as their own controls

(or when they are matched to very similar subjects)

For each subject we could have a pre and a post treatment score (e.g. pain, blood pressure, algometer, cholesterol levels, range of motion…)

The null hypothesis would be that the difference between pre and post scores would be 0 (treatment is not effective)

If the difference is sufficient, the null hypothesis can be rejected

T-Test #3 Independent t-TestAka independent groups t-TestMost commonly usedUsed when you have 2 groups

(2samples) out of an entire populationHo = X control = X treatment

Analysis of Variance (ANOVA)The t-Test only allows us to compare 2

groupsWhat if we have a study comparing 2 or

more types of treatment with a controls of both no treatment and placebo?

ANOVA is designed to handle multiple groups similar to what the t-Test does with 2 groups

Analysis of Covariance (ANCOVA)Sometimes studies nuisance variables

impact the dependent variable (outcome measures) but not the dependent variable (e.g. treatment).

These unwanted variables can interfere with our analysis of the data

Example…

Example We want to see if one of two treatment protocols will

have a positive effect upon patients with low back pain The patients are randomly assigned to the treatment

and control groups We realize from the histories that there are factors that

impact recovery from low back pain that we have not accounted for (e.g. obesity, smoking, occupation, age etc. etc.) These factors could impact rate of recovery (dependent variables)

The Analysis of Covariance pulls those possible confounding… nuisance factors out

Nonparametric TestsWilcoxon Signed Rank Test—Wilcoxon Ranked Sum Test—

equivalent to the Student t-testKruskal-Wallis Test– equivalent to the

one-way analysis of variance

Summary “Significance” is used in different ways…

statistical & medical The most commonly used inferential statistical

test is the Student t-Test (which has 3 versions depending what you are comparing) it only compares 2 group(s)/sample

Hypothesis can be directional or non-directional…

Critical value is established by the researcher BEFORE the study is started (.01, .05, .10)

Summary….contd.Parametric tests assume several things

related to a relatively normal distribution of data

Analysis of variance is used for comparing more than two variables

Analysis of Covariance is used to account for and remove the effects of nuisance variables

Next WEEKS Class