What you will learn

Primer on Statistics for Interventional

CardiologistsGiuseppe Sangiorgi, MD

Pierfrancesco Agostoni, MDGiuseppe Biondi-Zoccai, MD

What you will learn• Introduction• Basics• Descriptive statistics• Probability distributions• Inferential statistics• Finding differences in mean between two groups• Finding differences in mean between more than 2 groups• Linear regression and correlation for bivariate analysis • Analysis of categorical data (contingency tables)• Analysis of time-to-event data (survival analysis)• Advanced statistics at a glance• Conclusions and take home messages


Variables

nominal ordinal discrete continuous

orderedcategories

ranks counting measuring

Death: yes/noTLR: yes/no

TIMIflow

BMIBlood pressure

QCA data (MLD, late loss)

Stent diameterStent length

Types of variables

Radial/brachial/femoral

QUANTITYCATEGORY

Variables

discrete continuous

counting measuringBMI

Blood pressureQCA data (MLD, late loss)

Stent diameterStent length

Types of variables

QUANTITY

Variables

PAIRED OR

REPEATEDMEASURES

UNPAIREDOR

INDEPENDENTMEASURES

egblood pressure measuredtwice in the same patientsat different times

egblood pressure measuredin several different groups of patients only once

Types of

variables

Parametric and non-parametric tests

Whenever normal or Gaussian assumptions are valid, we can use PARAMETRIC tests, which are usually more sensitive and powerful

However, if an underlying normal cannot be safely assumed (ie there is non-gaussian distribution), NON-PARAMETRIC alternatives should be employed, as they are more robust and efficient

In some cases, albeit uncommonly in clinical cardiovascular research, there are alternatives to non-parametric tests in the presence of violations to normality assumptions

Such alternatives are mathematical transformations, such as the logarithmic (Ln), the power (^x), or the square root (√) trasformation

Alternatives to non-parametric tests

Alternatives to non-parametric tests

Categorical data: compare proportions in groups

Continuous data: compare means or medians in groups

Normal data; use t test

Non-normal data; use Mann Whitney

U testNormal data; use ANOVA

Non-normal data; use Kruskal Wallis

Are data categorical or continuous?

How many groups?

Two groups; normal data?

More than two groups; normal data?

Statistical tests

Variables

CATEGORY QUANTITY

nominal ordinal discrete continuousorderedcategori

es

ranks counting measuring

BinomialChi-square test

Fisher test

Chi-square testSign test

K-S test Kruskal-Wallis test

Mann-Whitney U test Spearman rho test

Wilcoxon test

Student’s t testMann-Whitney U test

Wilcoxon testAnalysis of VarianceKruskal-Wallis test

Kolmogodorov-Smirnov testLinear correlationLinear regressionSpearman rho test

Statistical tests

FreewaresEpi-info www.cdc.govRevMan www.cochrane.org

…or just go to www.google.com and search for the test you need…

Proprietary softwaresBMDPMinitabPrimerSASSPSSStataStatistica

Softwares

SPSS

SPSS

SPSS

SPSS

1) How can we compare EF in men vs. females after MI?

2) How does blood pressure change before and after therapy in a group of patients treated with B-blockers?

3) How can we test if there is difference in in-hospital death in patients treated with trombolysis vs. PCI?

Questions

4) How can we compare the occurrence of strokein AF patients treated with oral anticoagulant therapy vs.oral aspirin during a long term follow-up?

5) Can we predict discharge EF using peak CK values after MI?

Questions


What you will learn

• Finding differences in mean between two groups– independent groups (two-sample t-test)– dependent groups (paired t-test)– non-parametric alternatives: Mann-Whitney U

test (rank sum) and Wilcoxon test (signed rank)

Compare variables

Agostoni et al. AJC 2007

late loss (mm)

3,22,41,6,8-,0-,8

A

Freq

uenc

y

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Freq

uenc

y

50

40

30

20

10

0

Late loss in 2 different stents

Continuous or categorical variable?

Compare variables

late loss (mm)

3,22,41,6,8-,0-,8

A

Freq

uenc

y

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Freq

uenc

y

50

40

30

20

10

0


Compare continuous variables

Paired or unpaired data?

late loss (mm)

3,22,41,6,8-,0-,8

A

Fre

quen

cy

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Fre

quen

cy

50

40

30

20

10

0

Parametric or non-parametric test?

Compare continuous unpaired variables


late loss (mm)

3,22,41,6,8-,0-,8

A

Fre

quen

cy

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Fre

quen

cy

50

40

30

20

10

0

Mean: 0.45 SD: 0.76 Mean: 0.55 SD: 0.76

Compare variables

late loss (mm)

3,22,41,6,8-,0-,8

A

Fre

quen

cy

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Fre

quen

cy

50

40

30

20

10

0

Median: 0.29 IQR: -0.09–0.66 Median: 0.41 IQR: -0.02–0.85

Compare variables

late loss (mm)

3,22,41,6,8-,0-,8

A

Fre

quen

cy

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Fre

quen

cy

50

40

30

20

10

0

Mean: 0.45 SD: 0.76 Mean: 0.55 SD: 0.76


Compare variables

Value

Freq

uenc

y

Mean Mean

SD SD

Student t test for unpaired data: p=0.14Unpaired: same variable in different patients at same time

If parametric…

Mann Whitney U test for unpaired data: p=0.03

MedianMedian

IQRIQR

MedianMedian

IQRIQR

If non-parametric…Fr

eque

ncy

Value

Unpaired Student t test

Unpaired Student t testGroup Statistics

267 ,4533 ,75892 ,04645295 ,5468 ,76173 ,04435

typestentchypertaxus

late lossN Mean Std. Deviation

Std. ErrorMean

AB

Independent Samples Test

,002 ,962 -1,455 560 ,146 -,0935 ,06423 -,21964 ,03268

-1,456 554,860 ,146 -,0935 ,06422 -,21962 ,03266

Equal variancesassumedEqual variancesnot assumed

late lossF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

Student t test• A t-test is any statistical hypothesis test in which the

test statistic has a Student's t distribution if the null

hypothesis is true• It is applied when the population is assumed to be

normally distributed but the sample sizes are small

enough that the statistic on which inference is based is

not normally distributed because it relies on an uncertain

estimate of standard deviation rather than on a precisely

known value (if we knew it, we could use the Z-test)

Student t test• It is used to test the null hypothesis that the means of

two normally distributed populations are equal• Given two data sets (each with its mean, SD and number

of data points) the t test determines whether the means

are distinct, provided that the underlying distributions

can be assumed to be normal• the Student t test should be used if the variances (not

known) of the two populations are also assumed to be

equal; the form of the test used when this assumption is

dropped is sometimes called Welch's t test

Student t test

Mann Whitney rank sum U test

Mann Whitney rank sum U test

Ranks

267 266,65 71194,50295 294,94 87008,50562

typestentchypertaxusTotal

late lossN Mean Rank Sum of Ranks

A

B

Test Statisticsa

35416,50071194,500

-2,063,032

Mann-Whitney UWilcoxon WZAsymp. Sig. (2-tailed)

late loss

Grouping Variable: typestenta.

Ranking• The basic concept of non-parametric tests is ranking• The single values of the variable to analyze are not

evaluated according to their absolute value but to the

“rank” (or position) they assume in the merged

distributon of the values from lower to higher.

Driver Endeavor

17 2119 2119 2117 2118 6

6-17-17-18-19-19-21-21-21-21

1 2 3 4 5 6 7 8 9 10

Comparisonparametric/non-parametric tests

Robustness: non-parametric tests are much less likely

than the t tests to give a spuriously significant result

because of outliers – they are more robust

Efficiency: when normality holds, non-parametric tests

have an efficiency of about 95% when compared to

parametric tests. For distributions sufficiently far from

normal and for sufficiently large sample sizes, non-

parametric tests can be considerably more efficient

late loss (mm)

3,22,41,6,8-,0-,8

A

Fre

quen

cy

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Fre

quen

cy

50

40

30

20

10

0

Mean: 0.45 SD: 0.76 Mean: 0.55 SD: 0.76


Compare variables

Student t test for unpaired data: p=0.14

Mann Whitney U test for unpaired data: p=0.03

Late loss in restenotic lesions in two different stents

Mean: 1.75 SD: 0.51 Mean: 1.82 SD: 0.62

Unpaired Student t test: p=0.48


late loss (mm)

3,22,41,6,8-,0-,8

A

Fre

quen

cy

50

40

30

20

10

0

late loss (mm)

3,22,41,6,8-,0-,8

B

Fre

quen

cy

50

40

30

20

10

0

Late loss in non restenotic lesions in two different stents

Mean: 0.14 SD: 0.39 Mean: 0.27 SD: 0.44

Unpaired Student t test: p=0.002


Value

Freq

uenc

y

Paired: same variable in same group at different time

Paired Student t test

MAGIC, Lancet 2004

Significant increase in EF by paired t test P=0.005EF at baseline and FU in patients treated with BMC for MI

48.7% (8.3)48.7% (8.3)

55.1% (7.4)55.1% (7.4)

Only 11 patients !!!Only 11 patients !!!


Does MLD change from post-procedure to follow-upin a group of patients receiving a stent?


Paired Samples Test

,5024 ,76115 ,03211 ,4393 ,5655 15,648 561 ,000mld post-mld fuMean SD

Std.ErrorMean Lower Upper

95% CI of theDifference

Paired Differences

t dfSig.

(2-tailed)

Paired Samples Statistics

2,7770 562 ,48155 ,020312,2746 562 ,85787 ,03619

mld postmld fu

Pair1

Mean N Std. DeviationStd. Error

Mean

MLD fu MLD postDifference



Wilcoxon test: non-parametric comparison of 2 paired variables

Wilcoxon signed rank test

Wilcoxon signed rank testDescriptive Statistics

562 1,50 4,40 2,4400 2,7500 3,1000562 ,00 4,31 1,8700 2,4000 2,8400

mld postmld fu

N Minimum Maximum 25th 50th (Median) 75thPercentiles

Ranks

407a 322,51 131263,00153b 168,74 25817,00

2c

562

Negative RanksPositive RanksTiesTotal

mld fu - mld postN Mean Rank Sum of Ranks

mld fu < mld posta.

mld fu > mld postb.

mld fu = mld postc.

Test Statisticsb

-13,764a

,000ZAsymp. Sig. (2-tailed)

mld fu -mld post

Based on positive ranks.a.

Wilcoxon Signed Ranks Testb.


What you will learn

• Finding differences in mean between more than 2 groups– One-way analysis of variance– Non-parametric alternatives: Kruskal-Wallis,

Friedman

Value

Freq

uenc

y

Unpaired: same variable in >2 groups at same time

Three (or more) groups: what happens?

Always ask yourself… Paired or not? Parametric or not?


If ANOVA is significant, search where the difference is…

POST-HOC TESTSCAMELOT, JAMA 2004

1-way ANalysis Of VAriance

T test with P-value no more 0.05 but 0.05/n of tests performedin this case 0.01666

1-way ANOVA

1-way ANOVA• As with the t-test, ANOVA is appropriate when the data

are continuous, when the groups are assumed to have

similar variances, and when the data are normally

distributed• ANOVA is based upon a comparison of variance

attributable to the independent variable (variability

between groups or conditions) relative to the variance

within groups resulting from random chance. In fact,

the formula involves dividing the between-group

variance estimate by the within-group variance estimate

1-way ANOVADescriptives

blood pressure pre

5 90,0000 2,82843 1,26491 86,4880 93,5120 86,00 94,004 90,0000 4,08248 2,04124 83,5039 96,4961 85,00 95,004 90,0000 1,63299 ,81650 87,4015 92,5985 88,00 92,00

13 90,0000 2,73861 ,75955 88,3451 91,6549 85,00 95,00

placABTotal

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval forMean

Minimum Maximum

ANOVA

blood pressure pre

,000 2 ,000 ,000 1,00090,000 10 9,00090,000 12

Between GroupsWithin GroupsTotal

Sum ofSquares df Mean Square F Sig.


blood pressure post 1 month

5 90,0000 3,00000 1,34164 86,2750 93,7250 85,00 93,004 80,0000 4,08248 2,04124 73,5039 86,4961 75,00 85,004 85,0000 1,63299 ,81650 82,4015 87,5985 83,00 87,00

13 85,3846 5,14034 1,42567 82,2783 88,4909 75,00 93,00

placABTotal



Minimum Maximum

ANOVA


223,077 2 111,538 11,866 ,00294,000 10 9,400

317,077 12



Post-hoc testDescriptives


5 90,0000 3,00000 1,34164 86,2750 93,7250 85,00 93,004 80,0000 4,08248 2,04124 73,5039 86,4961 75,00 85,004 85,0000 1,63299 ,81650 82,4015 87,5985 83,00 87,00

13 85,3846 5,14034 1,42567 82,2783 88,4909 75,00 93,00

placABTotal



Minimum Maximum

Multiple Comparisons

Dependent Variable: blood pressure post 1 monthBonferroni

10,0000* 2,05670 ,002 4,0971 15,90295,0000 2,05670 ,106 -,9029 10,9029

-10,0000* 2,05670 ,002 -15,9029 -4,0971-5,0000 2,16795 ,131 -11,2222 1,2222-5,0000 2,05670 ,106 -10,9029 ,90295,0000 2,16795 ,131 -1,2222 11,2222

(J) drugABplacBplacA

(I) drugplac

A

B

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound95% Confidence Interval

The mean difference is significant at the .05 level.*.


blood pressure post 2 months

5 90,00 2,646 1,183 86,71 93,29 87 944 79,00 4,082 2,041 72,50 85,50 74 844 86,00 1,633 ,816 83,40 88,60 84 88

13 85,38 5,455 1,513 82,09 88,68 74 94

placABTotal



Minimum Maximum

ANOVA


271,077 2 135,538 15,760 ,00186,000 10 8,600

357,077 12





5 90,00 2,646 1,183 86,71 93,29 87 944 79,00 4,082 2,041 72,50 85,50 74 844 86,00 1,633 ,816 83,40 88,60 84 88

13 85,38 5,455 1,513 82,09 88,68 74 94

placABTotal



Minimum Maximum


Dependent Variable: blood pressure post 2 monthsBonferroni

11,00* 1,967 ,001 5,35 16,654,00 1,967 ,208 -1,65 9,65

-11,00* 1,967 ,001 -16,65 -5,35-7,00* 2,074 ,021 -12,95 -1,05-4,00 1,967 ,208 -9,65 1,657,00* 2,074 ,021 1,05 12,95


(I) drugplac

A

B

MeanDifference





5 90,00 2,646 1,183 86,71 93,29 87 944 79,00 4,082 2,041 72,50 85,50 74 844 86,00 1,633 ,816 83,40 88,60 84 88

13 85,38 5,455 1,513 82,09 88,68 74 94

placABTotal



Minimum Maximum


Dependent Variable: blood pressure post 2 monthsBonferroni

11,00* 1,967 ,001 5,35 16,654,00 1,967 ,208 -1,65 9,65

-11,00* 1,967 ,001 -16,65 -5,35-7,00* 2,074 ,021 -12,95 -1,05-4,00 1,967 ,208 -9,65 1,657,00* 2,074 ,021 1,05 12,95


(I) drugplac

A

B

MeanDifference



Kruskal Wallis test:

non parametric comparison

of >2 unpaired

continuous variables

Non parametric test

Kruskal Wallis test

Ranks

5 7,004 7,004 7,00

13

drugplacABTotal

blood pressure preN Mean Rank

Test Statisticsa,b

,0002

1,000

Chi-SquaredfAsymp. Sig.

bloodpressure pre

Kruskal Wallis Testa.

Grouping Variable: drugb.

Kruskal Wallis test

Ranks

5 10,504 3,134 6,50

13

drugplacABTotal

blood pressurepost 1 month

N Mean Rank

Test Statisticsa,b

8,3392

,015

Chi-SquaredfAsymp. Sig.

bloodpressure post

1 month

Kruskal Wallis Testa.

Grouping Variable: drugb.

Post-hoc analysis with

Mann Withney U and

Bonferroni correction

Value

Freq

uenc

y

Paired: same variable in same patients at >2 different moments

Three (or more) groups: what happens?

Always ask yourself… Paired or not? Parametric or not?


Three (or more) paired groupsAgain ask yourself… Parametric or not?

If parametric: ANOVA for repeated measuresin SPSS… in the General Linear Model

If non-parametric: Friedman test


Friedman test

Ranksa

2,00

1,90

2,10

blood pressure preblood pressurepost 1 monthblood pressurepost 2 months

Mean Rank

drug = placa.

Test Statisticsa,b

5,111

2,946

NChi-SquaredfAsymp. Sig.

Friedman Testa.

drug = placb.

Ranksa

3,00

2,00

1,00


Mean Rank

drug = Aa.

Test Statisticsa,b

48,000

2,018


Friedman Testa.

drug = Ab.

Ranksa

3,00

1,00

2,00


Mean Rank

drug = Ba.

Test Statisticsa,b

48,000

2,018


Friedman Testa.

drug = Bb.

Descriptive Statistics

5

5

5


N

drug = placa.

Descriptive Statisticsa

88,00 90,00 92,00

87,50 91,00 92,00

88,00 89,00 92,50

25th 50th (Median) 75thPercentiles

drug = placa.


95 86,25 90,00 93,75

85 76,25 80,00 83,75

84 75,25 79,00 82,75

Maximum 25th 50th (Median) 75thPercentiles

drug = Aa.


92 88,50 90,00 91,50

87 83,50 85,00 86,50

88 84,50 86,00 87,50

Maximum 25th 50th (Median) 75thPercentiles

drug = Ba.

• Actually, the analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables

• There are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment:

• One-way ANOVA is used to test for differences in ≥3 independent groups • One-way ANOVA for repeated measures is used when the subjects

undergo repeated measures; the same subjects are used for each treatment

• Factorial ANOVA (2-way ANOVA) is used to study the effects of two more treatment variables. The most commonly used type of factorial ANOVA is the 2×2 design, where there are two independent variables and each variable has two levels or distinct values

• When one wishes to test two or more independent groups subjecting the subjects to repeated measures, one may perform a factorial mixed-design ANOVA, in which one factor is a between subjects variable and the other is within subjects variable. This is a type of mixed effect model

• Multivariate ANOVA (MANOVA) - more than one dependent variable• Analysis of Covariance (ANCOVA) – ANOVA and regression/correlation


CAMELOT, JAMA 2004

A mixed-design ANOVA is used to test for differences between

independent groups whilst subjecting participants to repeated

measures. In a mixed-design ANOVA model, one factor is a between-

subjects variable (drug) and the other is within-subjects variable (BP)

2-way ANOVA

Thank you for your attention

For any correspondence: [email protected]

For further slides on these topics feel free to visit the metcardio.org website:

http://www.metcardio.org/slides.html

mailto:[email protected]



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What you will learn