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Short overview of statistical methods

Hein Stigum

Presentation, data and programs at:

http://folk.uio.no/heins/

courses

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Agenda• Concepts

• Bivariate analysis– Continuous symmetrical– Continuous skewed– Categorical

• Multivariable analysis– Linear regression– Logistic regression

Outcome variable decides analysis

CONCEPTS

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Precision and bias

• Measures of populations– precision - random error - statistics

– bias - systematic error - epidemiology

Truevalue

Estimate

Precision

Bias

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Precision: Estimation

Population Sample

valueTrue

Estimate

( | )

Estimate with confidence interval

95% confidence interval: 95% of repeated intervals will contain the true value

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Precision: Testing

Population Sample

2

1

groupvalueTrue

groupvalueTrue2Estimate

1Estimate

|

|

group 1

group 2

p-value=P(observing this difference or more, when the true difference is zero)

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Precision: Significance level

Birth weight, 500 newborn, observe difference

H0: boys=girls 10 gr p=0.90

50 gr p=0.40

100 gr p=0.10

130 gr p=0.04

150 gr p=0.02

Ha: boys≠girls

p<0.05Significance level

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Precision: Test situations

• 1 sample test• Weight =10

• 2 independent samples• Weight by sex

• K independent samples• Weight by age groups

• 2 dependent samples• Weight last year = Weight today

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Bias: DAGs

Egest age

Dbirth weight

C2parity

C1sex

Associations Bivariate (unadjusted)Causal effects Multivariable (adjusted)

Draw your assumptions before your conclusions

WHY USE GRAPHS?

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Problem example

• Lunch meals per week– Table of means (around 5 per week)

– Linear regression0

1020

3040

50P

erce

nt

1 2 3 4 5 6 7Lunch meals per week

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Problem example 2

• Iron level by sex– Both linear and logistic regression

– Opposite results

meangirls

meanboys0

.02

.04

.06

.08

75 129100 10490 110Irom level in bloodIron level in blood

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Datatypes

• Categorical data– Nominal: married/ single/ divorced

– Ordinal: small/ medium/ large

• Numerical data– Discrete: number of children

– Continuous: weight

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Data type

Normal data

MeansT-testLinear regression

MediansNon-par tests

Freq tableCross, ChisquareLogistic regression

CategoricalNumerical

Yes No

Outcome data type dictates type of analysis

BIVARIATE ANALYSIS 1Continuous symmetric outcome: Birth weight

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Distribution0

.000

2.0

004

.000

6.0

008

De

nsity

0 2000 4000 6000weight

0.0

002

.000

4.0

006

.000

8D

ens

ity2000 3000 4000 5000 6000

weight

kdensity weightdrop if weight<2000kdensity weight

0 2,000 4,000 6,000weight

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Central tendency and dispersion

Mean and standard deviation:

Mean with confidence interval:

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Compare groups, equal variance?

• Equal • Not equal

2 0 2 4 2 0 2 4

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2 independent samples

Are birth weights the same for boys and girls?

2000 3000 4000 5000 6000Birth weight

2000

3000

4000

5000

6000

Birt

h w

eigh

t

Boys Girlssex

Scatterplot Density plot

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2 independent samples test

ttest weight, by(sex) unequal unequal variancesttest var1==var2 paired test

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K independent samples

• Is birth weight the same over parity?

Scatterplot Density plot

2000

3000

4000

5000

6000

Birt

h w

eigh

t

0 1 2-7Parity

2000 3000 4000 5000 6000Birth weight

012-7

Parity:

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K independent samples test

equal means?

Equal variances?

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Continuous by continuous • Does birth weight depend on gestational age?

Scatterplot Scatterplot, outlier dropped

200

03

000

400

05

000

600

0B

irth

wei

ght

200 300 400 500 600 700Gestational age

200

03

000

400

05

000

Birt

h w

eig

ht

200 220 240 260 280 300Gestational age

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Continuous by continuous tests

• Cut gestational age up in groups, then use T-test or ANOVA

or

• Use linear regression with 1 covariate

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Test situations

• 1 sample test• ttest weight =10

• 2 independent samples• test weight, by(sex)

• K independent samples• oneway weight parity

• 2 dependent samples (Paired)• ttest weight_last_year == weight_today

BIVARIATE ANALYSIS 2Continuous skewed outcome: Number of sexual partners

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Distributionkdensity partners if partners<=50

25%50% 75% 95%0.0

2.0

4.0

6.0

8.1

11 4 9 20 50Partners

N=394

Distribution of number of lifetime partners

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Central tendency and dispersion

Median and percentiles:

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2 independent samples

Do males and females have the same number of partners?

Scatterplot Density plot

0 10 20 30 40 50partners

050

100

150

200

Par

tner

s

Males FemalesGender

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2 independent samples test

equal medians?

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K independent samples

Do partners vary with age?

Scatterplot Density plot

050

100

150

200

Par

tner

s

18-29 30-44 45-60agegr3

0 10 20 30 40 50partners

Age:18-2930-4445-60

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K independent samples test

equal medians?

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Table of descriptives

ProportionsNormal Skewed

DescriptivesCenter Mean Median pDispersion Standard deviation Fractiles

Confidence intervals for center estimatesStandard error se(mean) se(p)95% Confidence interval mean ± 2*se(mean) p ± 2*se(p)

Numerical data

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Table of tests

ProportionsNormal Skewed

1 sample One sample T-test Kolmogorov-Smirnov Binomial2 independent samples Independent sample T-test Mann-Whitney U Chi-squareK independent samples ANOVA Kruskal-Wallis Chi-square2 dependent samples Paired sample T-test Wilcoxon signed rank test Mc-Nemar (2x2)

Numerical data

Categorical ordered:

use nonparametric tests

If N is large:

may use parametric tests

Remarks: If unequal variance in ANOVA:

Use linear regression with robust variance estimation

BIVARIATE ANALYSIS 3Categorical outcome: Being bullied

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Frequency and proportionFrequency:

Proportion with CI:

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Proportion, confidence interval

proportion:

standard error:

confidence interval:

nx

p x=”disease”n=total number

)(2)(

)1()(

pseppCI

npp

pse

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Crosstables

equal proportions?

Are boys bullied as much as girls?

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Ordered categories, trend

Trend?

equal proportions?

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Table of tests

ProportionsNormal Skewed

1 sample One sample T-test Kolmogorov-Smirnov Binomial2 independent samples Independent sample T-test Mann-Whitney U Chi-squareK independent samples ANOVA Kruskal-Wallis Chi-square2 dependent samples Paired sample T-test Wilcoxon signed rank test Mc-Nemar (2x2)

Numerical data

Categorical ordered:

use nonparametric tests

If N is large:

may use parametric tests

Remarks: If unequal variance in ANOVA:

Use linear regression with robust variance estimation

MULTIVARIABLE ANALYSIS 1Continuous outcome: Linear regression, Birth weight

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Regression idea

residual error,e

xofeffect ,tcoefficienb

covariate =x

outcome=y

:model

1

10

exbby

covariate = x,x

:cofactorsmany with model

21

22110 exbxbby

2500

3000

3500

4000

4500

5000

birt

h w

eigh

t (gr

am

)

250 260 270 280 290 300 310gestational age (days)

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Model and assumptions

• Model

• Association measure1 = increase in y for one unit increase in x1

• Assumptions– Independent errors

– Linear effects

– Constant error variance

• Robustness– influence

),0(, 222110 Nxxy

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Workflow

• DAG

• Scatterplots

• Bivariate analysis

• Regression– Model estimation– Test of assumptions

• Independent errors• Linear effects• Constant error variance

– Robustness • Influence

Egest age

Dbirth weight

C2parity

C1sex

539

2000

3000

4000

5000

6000

birt

h w

eigh

t (gr

am

)

200 300 400 500 600 700gestational age (days)

Categorical covariates

• 2 categories– OK

• 3+ categories– Use “dummies”

• “Dummies” are 0/1 variables used to create contrasts• Want 3 categories for parity: 0, 1 and 2-7 children• Choose 0 as reference• Make dummies for the two other categories

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generate Parity1 = (parity==1) if parity<.

generate Parity2_7 = (parity>=2) if parity<.

Create meaningful constant

Expected birth weight at:

7_21

)(tirth weighExpected b

43210 ParityParitysexgest

yE

gr1925 0

gest= 0, sex=0, parity=0, not meaningful

gest=280, sex=1, parity=0gr35241280 210

Model estimation

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coeff 95% conf. Int.Birth weight at ref 3524.3Gestational age

per day 6.0 (3.9 , 8.2)Sex

Boy 0Girl -139.2 (-228.9 , -49.5)

Parity0 01 232.0 (130.6 , 333.5)2-7 226.0 (106.9 , 345)

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Test of assumptions

• Plot residuals versus predicted y– Independent

residuals?

– Linear effects?

– constant variance?

-100

0-5

000

500

1000

1500

Res

idua

ls

3200 3400 3600 3800 4000Linear prediction

Outlier not included

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Violations of assumptions• Dependent residuals

Use mixed models or GEE

• Non linear effectsAdd square term

• Non-constant varianceUse robust variance estimation

-1-.

50

.51

200 220 240 260 280 300gest

-2-1

01

2re

s

3400 3500 3600 3700 3800p

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Influence

Outlier

Regression with outlier

Regressionwithout outlier

2000

3000

4000

5000

6000

Birt

h w

eigt

h

200 300 400 500 600 700Gestational age

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Measures of influence

• Measure change in:– Predicted outcome

– Deviance

– Coefficients (beta)• Delta beta

Remove obs 1, see changeremove obs 2, see change

-.6

-.4

-.2

0.2

Influ

ence

1 2 10Id

Delta beta for gestational age

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539-10

-8-6

-4-2

0D

fbet

a ge

stC

280

2000 3000 4000 5000 6000weight

beta for gestational age= 6.04

If obs nr 539 is removed, beta will change from 6 to 16

Removing outlier

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coeff 95% conf. Int.Birth weight at ref 3524Gestational age

per day 6 (4 , 8)Sex

Boy 0Girl -139 (-229 , -49)

Parity0 01 232 (131 , 333)2-7 226 (107 , 345)

coeff 95% conf. Int.Birth weight at ref 3531Gestational age

per day 17 (13 , 20)Sex

Boy 0Girl -166 (-252 , -80)

Parity0 01 229 (132 , 326)2-7 225 (112 , 339)

Full model Outlier removed

One outlier affected two estimates Final model

MULTIVARIABLE ANALYSIS 2Binary outcome: Logistic regression, Being bullied

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Ordered categories and model

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Categories Regression model

2 Logistic

3-7 Ordinal logistic

>7 Linear (treat as interval)

Interval versus ordered scale:

Interval scale

Ordered scale

1 2 3

low medium high

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Logistic model and assumptions

• Association measure Odds ratio in y for 1 unit increase

in x1

• Assumptions– Independent errors

– Linear effects on the log odds scale

• Robustness– influence

11

eOR

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Being bullied

• We want the total effect of country on being bullied. – The risk of being bullied depends on age

and sex.

– The age and sex distribution may differ between countries.

• Should we adjust for age and sex?

Ecountry

Dbullied

C1age

C2sex

No, age and sex are mediating variables

N % p-value OR 95% conf. Int.Country <0.001

Sweden 407 8.7 1Island 448 10.9 1.3 (0.8 , 2)Norway 379 16.2 2.0 (1.3 , 3.2)Finland 409 25.9 3.7 (2.4 , 5.6)Denmark 436 23.4 3.2 (2.1 , 4.9)

Logistic: being bullied

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ORRR if outcome is rareOR>RR (further from 1) if the outcome is common

Prevalence of being bullied=17%

Roughly:Same risk of being bullied in Island as in Sweden.2 times the risk in Norwayas in Sweden.

3 times the risk in Finnlandas in Sweden.

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Summing up• DAGs

– State prior knowledge. Guide analysis

• Plots– Linearity, variance, outliers

• Bivariate analysis– Continuous symmetrical Mean, T-test, anova– Continuous skewed Median,

nonparametric– Categorical Freq, cross, chi-square

• Multivariable analysis– Continuous Linear regression– Binary Logistic regression