OBJECTIVES Understand the reason for and use of statistics Understand the reason for and use of statistics Review descriptive statistics Review descriptive statistics Measures of central tendency Measures of central tendency Measures of variability Measures of variability Measures of relationship Measures of relationship Inferential Statistics Inferential Statistics Parametric Parametric Non-parametric Non-parametric

Average: Mode: Median: Standard deviation 198.3

What are statistics for? Painting a mathematical picture….or simplifying large sets of data Painting a mathematical picture….or simplifying large sets of data Identifying if relationships exist Identifying if relationships exist Establishing the probability of a cause and effect relationship Establishing the probability of a cause and effect relationship

How many tests are there? There are well over 100 statistical calculations and tests There are well over 100 statistical calculations and tests However, many are rarely if ever used However, many are rarely if ever used Therefore, we will focus on those that are used most often… Therefore, we will focus on those that are used most often… Descriptive (Part I) and Inferential Statistics (Part II) Descriptive (Part I) and Inferential Statistics (Part II)

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. 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. “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. “Understanding t-test increases this access to 67 percent. “Familiarity with each additional test gradually increases the percentage of accessible articles.” “Familiarity with each additional test gradually increases the percentage of accessible articles.”

Descriptive Statistics*** Purpose --- To simply describe the data, the sample, or the population. Purpose --- To simply describe the data, the sample, or the population. Descriptive Statistics DO NOT establish a possible cause & effect relationship Descriptive Statistics DO NOT establish a possible cause & effect relationship

Inferential Statistics* Purpose --- In experimental research uses a sample of the population – inferential statistics permits the researcher to generalize from the sample data to the entire population. Purpose --- 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. 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 generally the preferred measure of central tendency Is used frequently for other calculations Is used frequently for other calculations

Measures of Variability (dispersion) Measures of Variability (dispersion)

Measures of Variability*** AKA spread. AKA spread. If scores are similar….they have low variability (homogeneous) If scores are similar….they have low variability (homogeneous) If scores are dissimilar…they have high variability (heterogeneous) If scores are dissimilar…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 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

Range** Simplest measure of variability Simplest measure of variability The difference between the lowest and highest score The difference between the lowest and highest score Usually reported in the literature as “range” (e.g. the range of ages was...) Usually reported in the literature as “range” (e.g. the range of ages was...) The range is an unstable calculation because it is only based upon 2 numbers The range is an unstable calculation because it is only based upon 2 numbers

Standard Deviation & Variance*** These measures are calculated based upon ALL data scores and are therefore better represent the data set These measures are calculated based upon ALL data scores and are therefore better represent the data set They are used with other statistical calculations They are used with other statistical calculations

Variance** Related to the amount that an individual score VARIES from the mean. Related to the amount that an individual score VARIES from the mean. This measurement is how much the scores deviate from the mean, BUT it uses all the scores. This measurement is how much the scores deviate from the mean, BUT it uses all the scores.

Standard Deviation*** Is the square root of the variance Is the square root of the variance How widely the values in a set are spread apart. How widely the values in a set are spread apart. A large SD tells you that the data are fairly diverse, while a small SD tells you the data are pretty tightly bunched together. A large SD tells you that the data are fairly diverse, while a small SD tells you the data are pretty tightly bunched together. If a mean is presented in the data, SD MUST be there at well. If a mean is presented in the data, SD MUST be there at well.

Standard Deviation (aka S.D.)***

Standard Deviation example*** If a group of people in an experiment had a mean age of 40, with a standard deviation of 4, what does this mean? If a group of people in an experiment had a mean age of 40, with a standard deviation of 4, what does this mean? 1. 68% of this group were between ages 36 and 44 (1 sd +/- the mean) 1. 68% of this group were between ages 36 and 44 (1 sd +/- the mean) 2. 95% were between 32 and % were between 32 and % were between 28 and % were between 28 and 52

Practice question Your treatment group (n:50) has a mean score on an IQ test of 100 with a standard deviation of 15. Your treatment group (n:50) has a mean score on an IQ test of 100 with a standard deviation of 15. How many subjects fall within one standard deviation of the mean? How many subjects fall within one standard deviation of the mean? If a subject has an IQ of 125, how many SD does he/she fall from the mean? If a subject has an IQ of 125, how many SD does he/she fall from the mean? If an individual has an IQ of 135, what percentage of people is he/she smarter than? If an individual has an IQ of 135, what percentage of people is he/she smarter than?

Correlations*** Measures of central tendency and variability describe only ONE variable Measures of central tendency and variability describe only ONE variable CORRELATIONS describe the relationship between TWO variables CORRELATIONS describe the relationship between TWO variables Correlations can be positive, negative or zero Correlations can be positive, negative or zero Correlations range from +1 through -1 Correlations range from +1 through -1

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

Summary Descriptive statistics do just that 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 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 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 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 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 Correlations are an important and often used type of descriptive statistics Descriptive Statistics must be well understood in order to understand inferential statistics Descriptive Statistics must be well understood in order to understand inferential statistics

Statistical & Medical Significance** It is important to keep in mind that statisticians use the word “significance” to represent the results of testing a hypothesis 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 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 & statistically significant Medically but not statistically significant Medically but not statistically significant Statistically but not medically significant Statistically but not medically significant Neither statistically or 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. 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.

Before the Experiment Researcher must determine the: Researcher must determine the: Critical value Critical value Directional or non-directional hypothesis Directional or non-directional hypothesis Power Power In order to determine the sample size. In order to determine the sample size.

Critical Value*** The researcher must establish the value at which they will consider the results “significant”…this is referred to as the 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 There is some subjective and somewhat arbritrary decision to be made in this regard by the researcher The customary minimal critical values are either p<.05 or p<.01 The customary minimal critical values are either p<.05 or p<.01

Power Determines the level of certainty that the results are accurate Determines the level of certainty that the results are accurate

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

Directional Hypothesis/Non… & Critical Value** These two decisions need to be made before the study is started and will determine to some degree if your results will be significant These two decisions need to be made before the study is started and will determine to some degree if your results will be significant …

Describing Results*** p values p values Confidence interval (CI) Confidence interval (CI) Odds ratios Odds ratios Number needed to treat (NNT). Number needed to treat (NNT).

p Values*** Most common form evaluating results, although not the best Most common form evaluating results, although not the best Tells you the odds of your result being caused by chance Tells you the odds of your result being caused by chance.05,.01.05,.01 Written as p<.05. Written as p<.05.

Confidence Interval*** Also give you the odds that the results are due to chance Also give you the odds that the results are due to chance Also give you the range in which the results should fall Also give you the range in which the results should fall Example: Significant improvement in the VAS, 95% CI (25-50). Example: Significant improvement in the VAS, 95% CI (25-50).

Odds Ratios*** It is the ratio of the number in the treatment group with improvement divided by the number without improvement compared to the number in the control group with improvement divided by the number without improvement It is the ratio of the number in the treatment group with improvement divided by the number without improvement compared to the number in the control group with improvement divided by the number without improvement 20/5 divided by 5/20= 16 20/5 divided by 5/20= 16 The higher the number the better the result The higher the number the better the result CI and odds ratios used together CI and odds ratios used together 95% CI odds ratio (2-8). 95% CI odds ratio (2-8).

NNT** Number (of patients) needed to treat Number (of patients) needed to treat This gives you the number of patients you need to treat before you have someone improve. The closer to 1 you get the better the treatment This gives you the number of patients you need to treat before you have someone improve. The closer to 1 you get the better the treatment

Post-operative pain 50%↓

Review of Descriptive Statistics

*** Mean Mean Median Median Mode Mode Variance Variance Standard Deviation Standard Deviation Odds ratio Odds ratio NNT NNT CI CI

*** Prior to designing experiment, effect of power, critical value, hypothesis and change measured Prior to designing experiment, effect of power, critical value, hypothesis and change measured Statistical and medical significance Statistical and medical significance

Standard Deviation (aka S.D.)

Inferential Statistics Parametric versus Nonparametric

Parametric Statistics 3 most commonly used tests & some related concepts commonly expressed in the literature

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

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” “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 Statistical Tests t-Test aka Student t-Test ANOVA Analysis of covariance

t- Test*** Developed by Gosset under the pseudonym Student Developed by Gosset under the pseudonym Student 3 different versions of the t-tests that apply to 3 different research designs 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 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 The larger the difference in the calculated t scores, the greater the chance that the null hypothesis can be rejected

t-Test*** 3 different general ways to use the student t-test 3 different general ways to use the student t-test 2 variations of each…dependent upon the type of hypothesis the researcher uses 2 variations of each…dependent upon the type of hypothesis the researcher uses The hypothesis can either be directional or non-directional The hypothesis can either be directional or non-directional

t-Test #1 Single Sample*** Compares the sample to the mean value of the population Compares the sample to the mean value of the population Is not used often because the mean for the population if usually not known 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 15 (although not everyone in the U.S. has had the test, enough have been tested to accept the data as representative E.g. Stanford Binet I.Q. test….has a mean of 100 and 1 S.D. of 15 (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) 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…) 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) 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 If the difference is sufficient, the null hypothesis can be rejected

T-Test #3 Independent t-Test*** AKA independent groups t-Test AKA independent groups t-Test Most commonly used Most commonly used Used when you have 2 groups (2 samples) out of an entire population Used when you have 2 groups (2 samples) out of an entire population Ho = X control = X treatment Ho = X control = X treatment

Analysis of Variance (ANOVA)*** The t-Test only allows us to compare 2 groups The t-Test only allows us to compare 2 groups What if we have a study comparing 2 or more types of treatment with a controls of both no treatment and placebo? What 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 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 independent variable (e.g. treatment). Sometimes studies nuisance variables impact the dependent variable (outcome measures) but not the independent variable (e.g. treatment). These unwanted variables can interfere with our analysis of the data These unwanted variables can interfere with our analysis of the data Example… Example…

Example*** We want to see if one of two treatment protocols will have a positive effect upon patients with low back pain 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 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) 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 The Analysis of Covariance pulls those possible confounding… nuisance factors out

Regression Analysis** Regression analysis falls somewhere between parametric and non-parametric statistics. Regression analysis falls somewhere between parametric and non-parametric statistics. When there are a number of variables, and there appears to be some trends, but nothing seems to be statistically significant, this measurement will be used. When there are a number of variables, and there appears to be some trends, but nothing seems to be statistically significant, this measurement will be used.

Regression, cont. Regression measures how well data fits a straight line. Regression measures how well data fits a straight line. The data is systematically analyzed, variable by variable, until only those variables that “fit” are left in the model. The data is systematically analyzed, variable by variable, until only those variables that “fit” are left in the model. Also referred to as least fit or best fit. Also referred to as least fit or best fit.

Non-parametric Statistics*** These tests are based on the MEDIAN rather than the MEAN of the sample. These tests are based on the MEDIAN rather than the MEAN of the sample. Parametric tests are generally performed FIRST, then non-parametric tests. Parametric tests are generally performed FIRST, then non-parametric tests. Non-parametric tests are performed when the expectation of normalcy is violated. Non-parametric tests are performed when the expectation of normalcy is violated.

Nonparametric Inferential Statistics --- no Assumptions** Makes no assumptions about the distribution of the data (distribution free) Makes 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. 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 Is less powerful…meaning that a greater difference (or change) needs to be present in the data before a significant difference can be detected

Nonparametric Tests*** Wilcoxon Signed Rank Test— equivalent to the t-test; data is multidirectional Wilcoxon Signed Rank Test— equivalent to the t-test; data is multidirectional Wilcoxon Ranked Sum Test—equivalent to the Student t-test Wilcoxon Ranked Sum Test—equivalent to the Student t-test Kruskal-Wallis Test– equivalent to the one-way analysis of variance Kruskal-Wallis Test– equivalent to the one-way analysis of variance

Kappa*** Commonly used in chiropractic & medical literature to convey the degree of int er examiner and/or int ra examiner reliability Commonly used in chiropractic & medical literature to convey the degree of int er examiner and/or int ra examiner reliability Interexaminer– two or more examiners checking/evaluating or testing for the same finding 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 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 Kappa calculates the degree of agreement between the first and second check/evaluation or test

Kappa values *** < 0 Poor agreement < 0 Poor agreement 0 — 0.20 Slight agreement 0 — 0.20 Slight agreement 0.21 — 0.40 Fair agreement 0.21 — 0.40 Fair agreement 0.41 — 0.60 Moderate agreement 0.41 — 0.60 Moderate agreement 0.61 — 0.80 Substantial agreement 0.61 — 0.80 Substantial agreement 0.81 — 1.00 Almost perfect agreement 0.81 — 1.00 Almost perfect agreement

Summary “Significance” is used in different ways…statistical & medical “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 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… Hypothesis can be directional or non- directional… Critical value, power and direction of hypothesis is established by the researcher BEFORE the study is started. Critical value, power and direction of hypothesis is established by the researcher BEFORE the study is started.

Summary….contd. Parametric tests assume several things related to a relatively normal distribution of data 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 variance is used for comparing more than two variables Analysis of Covariance is used to account for and remove the effects of nuisance variables Analysis of Covariance is used to account for and remove the effects of nuisance variables

Knowledge Comprehension and understanding of facts, truths or principles.