Inferential Statistics - hru.ge statistika.pdf · Inferential Statistics Predict and generalize results to the general population Analyze, test hypotheses, and answer research questions

Inferential Statistics

Predict and generalize

results to the general

population

Analyze, test

hypotheses, and answer

research questions

using math & logic

Allow the testing of

hypotheses using data

obtained from

probability and non-

probability samples

Used to draw

conclusions

1

What is Statistical Significance?

When Statistical Significance is found

It indicates that the observed relationship between

independent variable (predictor) and dependent

variable (outcome) from the data is likely to be

real, not by chance

Statistical Significance can be determined via

p-values

Confidence intervals

2

3

Statistical vs. Clinical Significance

A statistically significant result means that the

finding was unlikely to have occurred by chance.

Does not necessarily imply clinical relevance

Results may still be clinically relevant even when no

relationship appears from statistical analysis

Need to consider that factor in findings when

evaluating an article for use

4

What is Probability?

When conducting a statistical analysis, we have to ask

ourselves:

What are the chances of obtaining the same result from a study

that can be carried out many times under identical conditions?

The probability of an event is its long-run relative

frequency of occurring (0% to 100%) in repeated trials

and under similar conditions.

By chance, there is a risk that the characteristics of any

given sample may be different from those of the entire

population.


Represented by the p value

What it represents

The probability that a difference exists based

on the statistical test performed

5


Probability is based on the concept of

sampling error.

Even when samples are randomly selected,

there is always the possibility of some error

occurring.

Remember: The characteristics of any given

sample may be different from those of the

entire population.

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7

Probabilities on a Number Line

0% 100% 50% 25% 75%

0 1 0.5 0.75 0.25

Increasing Certainty

Expressed in Decimals

Expressed in Percentages

8

Statistical Significance and p-values

Frequently, research tries to show that the difference between groups did not happen by chance, but based on the result of the intervention being studied.

p =0.05 means: there is a 5% (1 in 20) chance that the difference actually occurred by

chance. The probability of committing a Type I error is 5%.

p =0.01 means: there is 1% (1 in 100) chance that the difference actually occurred by

chance. The probability of committing a Type I error is 1%.

p =0.001 means: there is 0.1% (1 in 1000) chance that the difference actually occurred by

chance. The probability of committing a Type I error is 0.1%.

9

p= 0.01

0 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 etc…. 1.0

0 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 100%

p= 0.05

Closer to 1.0 = more likely result due to chance

p-value Plotted on a Number Line

What do those mean?

Normal Distribution & p-value

p-values fall at some

point on the normal

distribution curve

The alpha value

determines where a

significant result will

fall

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11

p -Value Example – Not Significant

0

Reject

(p-Value = 0.0668) (a = 0.05)

p Value = 0.0668

a = 0.05

Do Not Reject Null Hypothesis

12

p -Value Example - Significant

0

Reject

(p-Value = 0.021) (a = 0.05)

p Value = 0.021

a = 0.05

Reject Null Hypothesis

i.e., There is a Statistically Significant Difference

13

Hypothesis Testing: The Why

Why do we do hypothesis testing? To determine:

How much of this effect is a result of chance

How strongly are these two variables associated

What is the effect of the intervention

Two hypothesis are tested

Research or scientific hypothesis

Null hypothesis

14

Hypothesis Testing: The What

Hypothesis is a statement about a research question.

Commonly hypotheses are statements about

A distribution parameter

Is the average birth weight greater than 5.5lbs?

Relationship between parameters

Is obesity rate higher among Blacks than Whites?

Relationship about probability distributions

Is smoking associated with CVD?

15

Null Hypothesis &

Alternative Hypothesis

Null Hypothesis (Statistical Representation: H0) Assumption that there is NO difference between groups

Alternate Hypothesis (Statistical Representation: HA) Statement that declares that there IS a difference between

groups

a.k.a. Study hypothesis

May be expressed as directional hypothesis e.g., the patient’s weight will decrease as a result of increased

exercise

16

Critical Region in the Sampling

Distribution for a One-Tailed

Polit et al, 2008

17

Critical Regions in the Sampling

Distribution for a Two-Tailed

18

Mosby items and derived items © 2009 by Mosby, Inc.

Level of Significance

Level of significance: alpha level (α)

Probability of making type I error

0.05 or less

Means that when you see the p value equal 0.05 or

less, the chance that a type I error occurred was

reduced.

The pre-determined cut off point at which you

will reject the null hypothesis

Level of Significance

Another way to think about it:

If the probability is = .01, this means that the

researcher is willing to accept the fact that if the

study was done 100 times, the decision to reject

the null hypothesis would be wrong 1 time out of

those 100 trials.

If alpha is 0.05, then the decision to reject the null

hypothesis would be wrong 5 times of those 100.

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20

Hypothesis Testing, p-values, & Alpha Levels

For any p-value less than 0.05 (i.e. between 0 and 0.05) you will reject the null hypothesis.

This means that a difference exists i.e., the result NOT due to chance.

For any p-value greater than 0.05 (i.e., between 0.05 and 1.0) you will not reject the null hypothesis.

This means that no difference exists, i.e, the result may be due to chance.

Example: p = 0.0668, α = 0.05 The researchers set the

cut-off at 0.05 ("alpha level = 0.05") but the p-value was 0.0668. Therefore the probability that a difference exist (i.e, that the result was due to chance) was 0.0668 -->

This p-value is greater than 0.05 therefore there is no difference and the researchers should accept the null hypothesis, i.e., the result may be due to chance.

21

0

Reject or Accept the Null?

p-value = 0.0668

a = 0.05

Example: Comparing Groups

The following slides provide examples of tables

from a research study that studied two different

types of arterial bypass surgery through a

randomized controlled trial. It also compared costs.

Link to actual study: http://jama.ama-

assn.org/content/291/15/1841.full.pdf+html?sid=3bae3b7d-0a0d-4152-babf-cbe9d6a2b0e4

22

http://jama.ama-assn.org/content/291/15/1841.full.pdf+html?sid=3bae3b7d-0a0d-4152-babf-cbe9d6a2b0e4











Baseline data

should NOT be

statistically

different

If the baseline

groups are

similar, then you

have more

ability to claim

that the result is

caused by the

intervention

Puskas et al., 2004, JAMA, 291, 1841-1849

23

Example: Comparing Groups

24

Mean Differences Results via p-value

Puskas et al., 2004, JAMA, 291, 1841-1849

25

Mean Difference Results About

Intervention Costs via p-value

Puskas et al., 2004, JAMA, 291, 1841-1849

Types of Statistical Errors

Sampling error

The tendency for statistics to fluctuate from one

sample to another

Type I

Type II

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27

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The Power Analysis:

A Prevention Intervention for Errors

(1- β) is called the power of the test (or the

statistical power for the test/study).

It is the probability to find the difference between

groups in a sample when there is a difference.

To reduce likelihood of error, sometimes the

researcher needs to perform a priori power analysis.

Confidence Intervals

Method to describe the precision of an estimate

95% CI indicates that you are 95% confident that the results would fall between the high and low if you repeated the study

Shown as a range in publications The more narrow the range between the numbers the more

precise the result

Example: 95% CI (A, B)

Example: 99% CI (C – D)

29

Confidence Intervals

To describe the precision of the estimate: Narrow confidence intervals = more precise

Wide confidence intervals = less precise

Two types of measurement level data work with CI when determining statistical significance Continuous

Discrete/Ratio

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31

Statistical Significance via Confidence

Intervals: Continuous Data

With continuous data, the differences between the means determines if the result is significant

If the 95% CI includes ‘0’, the result is not significant Example

Mean difference 5, 95% CI (-1.1, 8) Not Significant

Mean difference 5, 95% CI (2, 8) Significant

32



• Continuous results include mean differences, percent differences

and others.

• Mean difference is present if the CI is above or below ZERO,

but does not include it.

0 - 0.5 +0.5 - infinity + infinity

‘0’ indicates no difference

< 0 = Less than > 0 = More than

33



More about results with continuous data

Show outcome differences between groups

If the 95% CI includes ‘0’, the result is not significant 0 = no difference between groups

> 0 = outcome 1 is more likely in group 1 than in group 2

< 0 = outcome 1 is less likely in group 1 than in group 2

34

Statistical Significance via Confidence Intervals: Example – Cost Differences Between CABG Procedures

How it

appears

in a

journal

article

A B

C

35

Statistical Significance via Confidence Intervals: Example A – 1 Year Follow Up Cost Differences Between CABG Procedures

- infinity + infinity

95% CI

0 - 1000 -265 2192 -2643

Result: The One Year Follow-up Cost difference is minus $265

Interpretation: In a comparison between Off-pump vs Conventional CABG, the One Year

Follow-up Cost is on average 265 dollars less for Conventional CABG than for Off-Pump

CABG

Statistical Significance: Confidence interval DOES CROSS ‘0’, so the procedure is

sometimes more costly and sometimes less costly; therefore, NOT statistically significant.

36

Statistical Significance via Confidence Intervals: Example B – Total Acute Cost Differences Between CABG Procedures

0 - 1000 2272

- infinity

3732 755

+ infinity

95% CI

Result: The total acute cost difference is $2272

Interpretation: In a comparison between Off-pump vs Conventional CABG, the total acute

cost is on average 2272 dollars more for Conventional CABG than for Off-Pump CABG.

Statistical Significance: Confidence interval DOES NOT cross ‘0’, so conventional

CABG is always more costly; therefore YES, it is statistically significant.

37

Statistical Significance via Confidence Intervals: Example C – Professional Cost Differences Between CABG Procedures

95% CI

0 41 - infinity + infinity

- 300 385

Result: The professional cost difference is $41.

Interpretation: In a comparison between Off-pump vs Conventional CABG, the

“professional cost” is, on average, $41 more for Conventional CABG than for Off-Pump CABG

Statistical Significance?: Confidence interval CROSSES ‘0’, so the procedure is

sometimes more costly and sometimes less costly; therefore, NOT statistically

significant


Intervals: Discrete/Ratio Data

Discrete Data --> Ratios

If the 95% CI includes ‘1’, the result is not significant

Example Relative Risk Ratio 2, 95% CI (0.5, 6.4) Not Significant

Relative Risk Ratio 2, 95% CI (1.5, 6.4) Significant

Ratio Results

If the 95% CI includes ‘1’, the result is not significant 1 = equally as likely to occur

>1 = more likely to occur

<1 = less likely to occur

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Intervals: Discrete/Ratio Data

• If the result is a RATIO, plot the result on a number line that

crosses ONE (1)

HINT: you are looking for increased or decreased likelihood

1 0.5 2.0 0 + infinity

1:1 ratio indicates equally likely

< 1 = Less Likely to Occur > 1 = More Likely to Occur

40

Statistical Significance via Confidence Interval for Ratios: Example – Odds

Ratio for Developing Complications Post-CABG

A

41

Statistical Significance via Confidence Interval for Ratios:

Example A – Odds of Developing New A-Fib Post-CABG

1 0.33 0.68 0 + infinity

95% CI

1.4

Result: The Odds of developing new atrial fibrillation is 0.68

Interpretation: In a comparison between Off-pump vs Conventional CABG, patients are

32% less likely to develop new atrial fibrillation if they underwent Off-Pump CABG as

compared to Conventional CABG

Statistical Significance: Confidence interval CROSSES ‘1’, so it is sometimes more likely and sometimes less likely for patients to experience the complication; therefore, the result is

NOT statistically significant.


2

42

Statistical Significance via Confidence Interval for Ratios: Example B –

Odds of Readmission Post-CABG

B

43


Example B – Odds of Readmission Post-CABG

1 0.26 1.32 0

+ infinity

95% CI

3.35

Result: The odds of cardiac readmission is 1.32

Interpretation: In a comparison between Off-pump vs Conventional CABG, patients are

1.32 times more likely to have Cardiac Readmission if they underwent Off-Pump CABG as

compared to Conventional CABG

Statistical Significance: Confidence interval CROSSES ‘1’, so patients are sometimes

more likely and sometimes less likely to get readmitted; therefore, the result is NOT

statistically significant.


2


Interval for Ratios: Gauging Risk

Odds ratios can also determine the relative

risk for getting a complication or condition

Can help aid decision making by:

Healthcare professionals

Patients

44

45

PONV = Post Op Nausea & Vomiting

C


Example C – Risk for developing post-operative nausea with

ginger as a pre-op antiemetic (Chaiyakunapruk et al, 2006)

46


Example C – Risk for developing post-operative nausea with

ginger as a pre-op antiemetic (Chaiyakunapruk et al, 2006)

1 0.3 0.52 0 + infinity

95% CI

0.9

Result: The Relative Risk of developing post-op nausea & vomiting is 0.52

Interpretation: In a comparison between ginger vs no ginger pre-op, patients are

48% less likely to develop post op nausea & vomiting if they took ginger as

compared to if they did not take ginger.

Statistical Significance: Confidence interval DOES NOT cross ‘1’, so ginger

always prevents PONV; therefore YES, it is statistically significant.


2


Interval for Ratios: The Forest Plot

A Forest Plot is:

A graphical display designed to illustrate the

relative strength of treatment effects in multiple

quantitative scientific studies addressing the same

question

Frequently used with meta-analyses

47


Interval for Ratios: The Forest Plot

The left-hand column lists the names of the studies

The right-hand column is a plot of the measure of

effect (e.g. an odds ratio) for each of these studies

(often represented by a square) incorporating

confidence intervals represented by horizontal lines.

The area of each square is proportional to the study's

weight in the meta-analysis.

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Example: Forest Plot from a Meta-Analysis

(Systematic Review) of Ginger as Pre-Op Antiemetic

Chaiyakunapruk

et al. (2006)

American

Journal of

Obstetrics and

Gynecology,

194(1), 95-99.

This was the only non-significant

result. Can you describe why?

50

Caveat

Statistical significance does not mean clinical

significance and vice versa

Size of the p-value does NOT indicate importance of the

result

If statistically significant, results might not be practical

e.g., Treatment could be effective, but might involve costly or

inaccessible procedure

Even if not statistically significant, results might be very

important

e.g., if sample size were increased, might see statistical

significance

Inferential Statistics

Tests of Difference(s)

Evaluating the degree of the relationship

between variables

51

Tests of Difference

t test Checks whether the

means of two groups

are different.

Useful for experimental

studies, but may be

applied to other studies

where two groups are

compared

52

Testing Difference through Variation

Analysis of variance (ANOVA)

Tests whether more than two group means differ

Considers the variation between groups and within groups.

Analysis of variance (ANOVA)

Tests whether more than two group means differ

Considers the variation between groups and within groups.

Multiple analysis of variance (MANOVA)

Also is used to determine differences in group means, but when there

is more than one dependent variable.

53

Tests of Difference

Chi-square (2)

Uses nominal data to determine whether groups

are different.

A nonparametric statistic used to determine

whether the frequency in each category is

different from what would be expected by

chance.

54

Correlation (r)

The correlation coefficient is a simple

descriptive statistic that measures the strength

of the linear relationship between two

interval- or ratio-scale variables (as opposed

to categorical, or nominal-scale variables), as

might be visualized in a scatter plot.

55

http://geography.uoregon.edu/bartlein/courses/geog417/images/measure.htm







Tests of Relationship

Most common test of correlation: Pearson

product-moment correlation Correlation coefficients can range in value from -1.0 to

+1.0 and also can be zero.

A zero coefficient means that there is no relationship

between the variables.

A perfect positive correlation is indicated by a +1.0

coefficient.

A perfect negative correlation by a -1.0 coefficient.

56

Scatter Plots: Correlation

57

Caution with Correlation

Correlation and Causality

One common mistake made by people

interpreting a correlational coefficient refers to

causality.

Example: When we see that depression and low

self-esteem are negatively correlated, we often

surmise that depression must therefore cause the

decrease in self-esteem.

58

Evaluating the Degree of Relationships

Between Variables: Regression

A correlation tells you IF a relationship exists

between variables

A regression tells you:

How strong a relationship is between the dependent and

two or more independent (explanatory) variables

How much of the dependent variable the independent

variable (or combination of variables) explains or predicts

How the dependent variable changes when the

independent variable is present or not

59

Common Types of Regression Analyses

Used in Health Research

Linear regression (also multiple linear regression)

Uses ordinal or ratio data

Logistic regression

Uses categorical (nominal) data

Frequently seen with odds ratios

Regression results are represented by r2 and look like

the example below in publications

Example: r2 = .7 (p=.0001)

60

Regression Graphs Look Like This

The dots show where X

and Y are linked and have

a relationship

The line is the regression

line, showing the direction

of the relationship

(predictive)

Closer the dots to the line =

stronger relationship,

stronger predictions

61

Regression Tables Look Like This

Average education and the

kinds of hospitals where

nurses could work were

significant predictors of the

nurse-to-population ratio

Those two variables

combined explain 70% of a

nurse-to-population ratio in

a state in Mexico.

62

Variables Coefficient p

value

Median Income 0.0001 .582

Average

Education 0.6000 .000

# Children/Family 0.104 .662

Total Nursing

Schools -.0004 .410

Ratio of Public:

Private Hospitals -1.685 .001

Squires & Beltran-Sánchez (2009): This study sought to investigate macro-level

socioeconomic predictors of the nurse-to-

population ratio in Mexico.

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Inferential Statistics - hru.ge statistika.pdf · Inferential Statistics Predict and generalize results to the general population Analyze, test hypotheses, and answer research questions