statistical methods in research - oluwadiyaoluwadiya.com/documents/workshops/advanced statistical...

Effective Use of Advanced

Statistical Methods in Research

OLUWADIYA KS

Professor of Surgery

Ekiti State University, Ado-Ekiti

www.oluwadiya.com

Objectives

1. Data transformation

2. Limitations of P-value

3. Statistics for comparing 2 or more groups with continuous data

4. Regression and Correlation

5. Risk Ratios and Odds Ratios

6. Survival Analysis

7. Sensitivity, Specificity and ROC Curves

8. Finding the right test for specific data

9. Introduction to Endnote

2

Why Transform Data?

The assumptions of most parametric methods:

Homogeneity of variance (Homoscedasticity)

Normality

Linearity

Data transformation is used to make your data

conform to the assumptions of the statistical

methods

Skewed data

Homoscedasticity and Normality

The data deviates from both homoscedasticity and normality.

Homoscedasticity and Normality

This data are both normal and have equal variance

Won’t it be nice if we would make the previous data look this way?

• We should always check the assumptions

that data follow a normal distribution with

uniform variance:

i. If the data meet the assumptions we can

analyze the raw data as described.

ii. If they are not met, we have two

possible strategies:

When to transform data?

1-We can use a method which does not

require these assumptions, such as a rank-

based (non-parametric) method.

2-We can transform the data mathematically

to make them fit the assumptions more

closely before analysis.

Methods of data transformation

In healthcare research, there are three commonly

used transformations for quantitative data:

1. Logarithmic transformation,

2. Square root transformation

3. Inverse (reciprocal) transformation.

Determining normality of data

Graphical method

Histograms and Normality plots

Boxplots

Normal Q-Q plots and detrended Q-Q plots

Statistical method

Skewness and kurtosis

Smolgrov-Smirnov statistics

10


11

Which of the two variables

has a normal distribution?


Age PCV

12


AGE PCV

13


AGE PCV

14


15

Normalizing data

We know that Age is

not normally

distributed

Log transformation:

use SPSS Compute

Sub menu (transform-

compute):

16

Normalize Age

The transformed variable

(log_age) which we asked

SPSS to create has been

created

17

Normalize Age: Result

Original Data (Age) Transformed data

18

Normalize PCV: Result

Log_Age Age

19

The problem with P

P values provide less information than confidence intervals.

Statistical significance tells us whether there is a difference and not how much difference there is.

A P value provides only a probability that an estimate is due to chance

A P value could be statistically significant but of limited clinical significance.

o A very large study might find that a difference of 0.5mmHg in BP between 2 rx groups is statistically significant but is this clinically relevant?

“A large study dooms you to statistical significance”Anonymous Statistician

20

Statistical tests in inferential statistics are designed to answer the

question ‘‘how likely is the difference found in a sample due to chance

(when actually no such difference exists in the population, the null-

hypothesis)?’’.

This is the only purpose they serve—the calculation of a

probability value.

They do not indicate clinical significance!

21

The problem with P

The clinical significance of a research finding – the extent

to which it may influence clinical practice – depends on

many factors.

I. Many of these factors are related to study design.

II. are the adverse effects of a treatment also studied in addition to

benefits?

III. Are the outcome measures clinically relevant (e.g., improvement

in symptoms and functioning rather than cognitive distortions)?

IV. Are the effects lasting?

V. Is the cost of treatment worth the effect and

VI. Can the study findings be generalized to patients across social

and clinical settings?

22

Clinical significance

Effects size (Cohen’s d)

Odds ratio

Absolute Risk Reduction (ARR)

Numbers needed to treat (NNT): this is the

reciprocal of ARR

23

Magnitude and clinical significance

Magnitude and clinical significance

No

response

response Total

Placebo A(60) B(40) A+B(100)

Drug C(40) D(60) A+B(100)

24

Risk of response on

antidepressant=0.6

Risk of response on

placebo = 0.4

Absolute risk reduction =

0.2

Number Needed to Treat

(NNT) = 1/0.2 = 5

Statistical Tests

Parametric tests

Continuous data; normally distributed

Non-parametric tests

Continuous data; not normally distributed

(Categorical or Ordinal data)

25

Comparison of two sample mean

Student’s T test

Assumes normally distributed continuous data.

No need to do the math, commonly generated by most statistics software

But…

Understand the underlying theory and assumption

26

Paired T-Test

Whereas T-test assumes there is independence of observations

Related samples i.e.

Paired T-test is meant for “before” and “after” studies

27

Used to determine if two or more samples are

from the same population i.e. no significant

difference between their means

Requires that….

Dependent variable is continuous data

Independent variable is categorical data

Independent variable = Grouping variable = Factor

The variable will consist of a number of categories or levels

There will be 2 or more of such categories or levels.

If there are only 2 categories, then the result will be identical to t-test

. 28

ANalysis Of VAriance (ANOVA)

ANalysis Of VAriance

An example

You measure the PCV of 50 patients who has sustained fractures.

Does the number of bones fractured affect the PCV of the patients ?

number of bone fractured is the independent variable or factor

0 bone fractured is one level of the factor

1 bone fractured is one level of the factor

2 bones fractured is one level of the factor

3 bones fractured is one level of the factor

PCV is the dependent variable

ANOVA in SPSS

Significant result…now what?

Yes oo! & there is a

significant

difference among

the means. I’m so

happy

Don’t

know

Na wa o

Ok

then

There are

more than 2

means

But is it

among all

means, or

just two?

Better do a

post hoc test

my friend.

What is that?

Better do it

now

Post Hoc tests

After the Fact comparisons of means used

to identify which specific pairs of means are

significantly different

Designed to reduce errors regardless of

how many pairs of means are compared

Post Hoc tests

Also called follow-up tests

Should be computed only after a significant ANOVA

They are like a collection of little t-tests

But they control overall type 1 error comparatively well

They do not have as much power as the omnibus test (the ANOVA) – so you might get a significant ANOVA & no sig. Follow-up

Purpose is to identify the locus of the effect (what means are different, exactly?)

Post Hoc tests

Post Hoc tests: Homogeneous subset

Correlation

Assesses the linear relationship between two variables Example: height and weight

Strength of the association is described by a correlation coefficient- r

r = 0 - .2 low, probably meaningless

r = .2 - .4 low, possible importance

r = .4 - .6 moderate correlation

r = .6 - .8 high correlation

r = .8 - 1 very high correlation

Can be positive or negative

Pearson’s or Spearman’s correlation coefficient

Tells nothing about causation

36

Correlation

Positive Correlation Negative Correlation

37

Regression

Based on fitting a line to data

Provides a regression coefficient, which is the slope of the line

o For example; y = 0 + 1x (simple linear regression)

Use to predict a dependent variable’s value based on the value of an independent variable.

E.g. In analysis of height and weight, for a known height, one can predict weight.

Much more useful than correlation

Allows prediction of values of Y rather than just whether there is a relationship between two variable.

38

Regression

Types of regression

Linear and Multiple - uses continuous data to predict

continuous data outcome

Logistic- uses continuous data to predict probability of

a dichotomous outcome

Poisson regression- time between rare events.

Cox proportional hazards regression- survival analysis.

39

40

• The simple linear regression equation is:

y = 0 + 1x

• Graph of the regression equation is a straight line.

• 0 is the intercept of the regression line on the yaxis.

• 1 is the slope of the regression line.

• y is the expected value of y for a given x value.

Simple Linear Regression Equation

41


Positive Linear Relationship

y

x

Slope 1

is positive

Regression line

Intercept0

42


Negative Linear Relationship

Slope 1

is negative

E(y)

x

Regression line

Intercept0

Slope 1

is negative

43


No Relationship

x

E(y)

Slope 1

is 0

Regression line

Intercept0

Risk Ratios

Risk is the probability that an event will happen.

Number of events divided by the number of people at risk.

Risks are compared by creating a ratio

Example: risk of colon cancer in those exposed to a factor vs. those unexposed

Risk Ratios

Typically used in cohort studies

Prospective observational studies comparing groups with various exposures.

Allows exploration of the probability that certain factors are associated with outcomes of interest

For example: association of smoking with lung cancer

Usually require large and long-term studies to determine risks and risk ratios.

45

Interpreting Risk Ratios

A risk ratio of 1 equals no increased risk

A risk ratio of greater than 1 indicates increased risk

A risk ratio of less than 1 indicates decreased risk

95% confidence intervals are usually presented

Must not include 1 for the estimate to be statistically significant.

Example: Risk ratio of 3.1 (95% CI 0.97- 9.41) includes 1, thus would not be statistically significant.

46

Odds Ratios

Odds of an event occurring divided by the

odds of the event not occurring.

Odds are calculated by the number of times an

event happens by the number of times it does

not happen.

o Odds of heads vs. the odds of tails is 1:1 or 1.

47

Odds Ratios

Are calculated from case control studies

Case control: patients with a condition (often rare) are compared to a group of selected controls for exposure to one or more potential etiologic factors.

Cannot calculate risk from these studies as that requires the observation of the natural occurrence of an event over time in exposed and unexposed patients (prospective cohort study).

Instead we can calculate the odds for each group.

48

Comparing Risk and Odds Ratios

For rare events, ratios very similar If 5 of 100 people have a complication:

o The odds are 5/95 or .0526.

o The risk is 5/100 or .05.

If more common events, ratios begin to differ If 30 of 100 people have a complication:

The odds are 30/70 or .43

The risk is 30/100 or .30

Very common events, ratios very different Male versus female births

The odds are .5/.5 or 1

The risk is .5/1 or .5

49

Risk reduction

Absolute risk reduction: amount by which risk is reduced.

Relative risk reduction: proportion or percentage reduction.

Example:

Death rate without treatment: 10 per 1000

Death rate with treatment: 5 per 1000

ARR = 5 per 1000

RRR = 50%

50

What is survival analysis?

Survival analysis is form of regression technique which models time to an event (death, recurrence, recover).

Unlike linear regression, survival analysis has a dichotomous (binary) outcome

Unlike logistic regression, survival analysis analyzes the time to an event

Able to account for censoring

Can compare survival between groups

Assessses relationship between covariates (variables) and survival time

defined event of interest (such as death, recurrence, new primary)

specify start and end time of study’s observation period

determine time to event or censoring for each subject

Survival analysis requires

† death from specific cancer = event

? lost to follow-up = censor

alive at last visit = censor

Survival Data

?

†

End of follow upStart

What is Censored data?

Patients who do not reach the event by the end of

the study or who are lost to follow-up.

Types of censored data include:

Event did not occur by the end of the study

Subject died from an unrelated cause

Subject lost to follow-up

Subject withdrew from study

54

Regression vs. Survival Analysis

Technique Mathematical

model

Yields

Linear

Regression

Y=B1X + Bo

(linear)

Linear changes

Logistic

Regression

Ln(P/1-P)=B1X+Bo

(sigmoidal prob.) Odds ratios

Survival

Analyses

h(t) =

ho(t)exp(B1X+Bo)

Hazard rates

When to use survival analysis

Estimate time-to-event for a group of individuals,

such as time until second heart attack for a group of

MI patients.

To compare time-to-event between two or more

groups, such as treated vs. placebo patients in a

randomized controlled trial.

To assess the relationship of co-variables to time-

to-event, such as: does weight, smoking, or

cholesterol influence survival time of MI patients?

Kaplan-Meier survival curves

Accounts for censoring

Provides a graphical means of comparing the

outcomes of two groups that vary by intervention

or other factor.

Survival rates can be measured directly from curve.

Difference between curves can be tested for

statistical significance.

Does not account for confounding or effect

modification by other covariates

Days

Surv

ival P

robabili

ty

Males

Female

Survival after Surgery (Kaplan-Meier curves)

Ticks = censor

Step = event

Cox Regression Model

Also called Proportional Hazards Survival Model.

Used to investigate relationship between an event (death,

recurrence) occurring over time and possible explanatory

factors (Covariates).

Reported result: Hazard ratio (HR).

Ratio of the hazard in one group divided by the hazard in another.

Interpreted same as risk ratios and odds ratios

HR 1 = no effect

HR > 1 increased risk

HR < 1 decreased risk

59

Cox Regression Model

Common use in long-term studies where

various factors called covariates might

predispose to an event.

Example: after radiotherapy, which factors

(age, race, tumor staging, etc.) might make

recurrence more likely.

60

Sensitivity

61

Sensitivity = A/(A+C)

Ability of a test to identify

correctly, individuals who are

affected

Proportion of people testing

positive

among affected individuals

Specificity

62

Sensitivity = D/(D+B)

Ability of a test to identify

correctly, individuals who are

not affected

Proportion of people testing

negative among non-affected

individuals

TN

Sp =

TN + FP

TP

Se =

TP + FN

Disease

Test

FP

TN

TP

Performance of a test

FN

NoYes

+

-

63

0 5 10 15 20

Quantitative result of the test

Distribution of quantitative test results among

affected and non-affected people (ideal case)

TN

Non affected:

Affected:

TP

Nu

mb

er o

f p

eople

tes

ted

Threshold for

positive result

64

0 5 10 15 20

TN TP

FN FP

Distribution of quantitative results among affected

and non-affected people (real life)

Non-affected:Threshold for

positive result


Nu

mb

er o

f p

eop

le t

este

d

Affected:

65

TNTP

FN

FP

Non affected:

Affected:Threshold for

positive result

Effect of Decreasing the Threshold

Nu

mb

er o

f p

eop

le t

este

d


0 5 10 15 20

66

TP

Se =

TP + FN

TN

Sp =

TN + FP

Effect of decreasing the Threshold

Disease

Test

FP

TN

TP

FN

NoYes

+

-

67

0 5 10 15 20

TNTP

FN

FP

Non-affected:

Affected:Threshold for

positive result

Nu

mb

er o

f p

eop

le t

este

d


Effect of Increasing the Threshold

68

TP

Se =

TP + FN

TN

Sp =

TN + FP

Effect of Increasing the Threshold

Disease

Test

FP

TN

TP

FN

NoYes

+

-

69

Performance of a Test and Threshold

Sensitivity and specificity vary in opposite

directions when changing the threshold

The choice of a threshold is a compromise

to best reach the objectives of the test What are the consequences of having false positives?

What are the consequences of having false negatives?

70

When false diagnosis (FP)

is worse than missed diagnosis (FN)

Example: Screening for congenital

toxoplasmosis

One should minimise false positives

Prioritise SPECIFICITY

71

When missed diagnosis (FN)

is worse than false diagnosis (FP)

Example: Testing for Helicobacter pylori

infection

One should minimise the false negatives

Prioritise SENSITIVITY

72

Receiver Operating Characteristics curve

(ROC curve)

Representation of relationship

between sensitivity and specificity for a test

Simple tool to:

define best cut-off value of a test

compare performance of two tests

73

ROC Curve 1

0 20 40 60 80 100

100

80

60

40

20

0

100-Specificity

Sensitiv

ity gcs

KTS

RTS

Triage RTS

74

ROC Curve

Score AUC 95% C. I Optimal cutoff

point

Cutoff sensitive/specific

Sensitivity

%

Specificity %

GCS 0.880 0.825 to 0.

923

9 100 68.4

tRTS 0.881 0.825 to 0.

924

9 100 54

RTS 0.883 0.827 to 0.

925

5.7 83.3 83.3

KTS 0.914 0.864 to 0.

950

12 100 70.7

75

76

Determining the Test (I)

What kind of variables are they?

1. Numerical variable

2. Ordinal variable

3. Categorical variable (Nominal)

How many groups are there?

T-test vs ANOVA

77

Determining the Test (II)

Are they “normal distribution”?

Parametric vs. nonparametric methods.

T-test vs. Mann-Whitney U test

ANOVA vs. Kruskal-Wallis test

78

Determining the Test (III)

Measurements are taken from the same

patient for more than one time (before

and after treatment); you should use

Paired t-test

Repeat-measures ANOVA

79

Determining the Test (IV)

Usually, data are analyzed after they are

completed (all the measurements are

finished); but there are some studies that

data input are still ongoing while the

researcher have started their analysis.

For these data; do survival analysis

Selecting the appropriate procedure among

the common statistical procedures

80

Independent Variable

Dependent Variable

Categorical Continuous

Categorical Chi Square Logistic regression

t-tests One-way ANOVA

Continuous Logistic regression Correlation Linear regression

For a more complete table, please see the book:

Getting to know SPSS. Second Edition

by Oluwadiya Kehinde

Further studies?

Getting to know SPSS.

Second Edition by

Oluwadiya Kehinde

[email protected]

www.oluwadiya.sitesled.com

81

mailto:[email protected]

Final thoughts…….

82

When reading a journal article…….

Keep In Mind That

No study is perfect

All data is dirty is some way or another;

research is what you do with that dirty data

Measurement involves making choices

Be Critical About Numbers

Every statistic is a way of summarizing complex

information into relatively simple numbers.

How did the researchers arrive at these numbers?

Who produced the numbers and what is their bias?

How were key terms be defined & in how many

different ways?


How was the choice for the measurement made?

What type of sample was gathered & how does that affect result?

Is the statistical result interpreted correctly?

If comparisons are made, were they appropriate?

Are there competing statistics?


With one foot in a

bucket of ice water,

and one foot in a

bucket of boiling

water, you are, on

the average,

comfortable.

Be critical about numbers:

Bias and Error

Thanks for your attention

88

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