overview of inferential statistics - interfetpthailand chavez/inferential... · 2 topics • bias...

1

Overview of Inferential Statistics

Jaranit Kaewkungwal, Ph.D.Faculty of Tropical Medicine

Mahidol University

2

Topics

• Bias & Chance

• Statistical Methods• Descriptive Statistics• Inferential Statistics

• Parameter Estimation• Hypothesis Testing

• Comparisons• Associations• Multi-variable Data Analysis

• Choosing the right statistical procedures

3

I

Bias & Chance

4

Bias Chance Truth

Possible Explanations of Clinical Outcomes

Bias vs.Chance

Observed = Truth + Error

Systematic error + Random error(Bias) (Chance)

5

Bias vs.ChanceObserved = Truth + Error

Systematic error + Random error

syst

olic

agegrp1 3

87

190

<=20 21−40 >= 41age group

τ

ε

τ

ε

Example of Statistical Analysis:

Analysis of variance:F = σ2

(τ+ε) /σ2ε

1890-1962

μ

6


Regression Analysis:

Y = Y + ε

Y = α + β1 X1Cigarette Consumption per Adult per Day

12108642

CH

D M

orta

lity

per 1

0,00

0 30

20

10

0



(Y = 11, X = 6)

εY=13, X = 6

7


Chi-Square test:χ2 = Σ(Ο−Ε)2/Ε

Kappa Statistics:Observed agreement - Expected Agreement

κ = Obs Agreemt - Expct Agreemt1 - Expected Agreement



100554554522No

46343YesMD#2NoYes

MD #1OBSERVED

10055455429.724.3No

4625.320.7YesMD#2

NoYes

MD #1EXPECTED

8

Bias:• A process at any stage of inference tending to produce results that depart systematically from the true values.

Bias vs.Chance

Chance:• The divergence of an observation on a sample from the true population value in either direction.• The divergence due to chance alone is called random variation

Bias and chance- are not mutually exclusive.

9

Bias vs.Chance

“A well designed, carefully executed study usually gives results that are obvious without a formal analysis and if there are substantial flaws in design or execution a formal analysis will not help.”

Johnson AF. Beneath the technological fix. J Chron Dis 1985 (38), 957-961

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Chance

“Free kick”

Prob

abili

ty o

f bei

ng h

it

50% 50%68%95%

2.5 %2.5 %

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Chance

“Free kick”

Prob

abili

ty o

f get

ting

goal

50% 50%68%95%

2.5 %2.5 %

12

Chance in Statistics: Descriptive Statistics

Standard Score

Raw Score

20 25 30 35 40

X = 30;SD = 5

13

Confidence Limits of μ : X + Zα/2,ν SX

X =SD = σ = 0.5 n=16

Upper LimitLower Limit

Chance in Statistics: Parameter Estimation

95% CI

μ = 80 (79.755−80.245)

μ X

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Chance in Statistics: Hypothesis Testing

Not Reject Ho !!μ1 = μ2

Ho: μ1 = μ2Ho: μ1 − μ2 = 0Ha: μ1 − μ2 = 0

μ1 μ2

15

Chance in Statistics: Hypothesis Testing

μ1 μ2

Ho: μ1 − μ2 = 0Ha: μ1 − μ2 = 0

Reject Ho !!μ1 < μ2

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II

Statistical Methods

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What is Statistics?

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Types of Statistics

• By Level of Generalization– Descriptive Statistics– Inferential Statistics

• Parameter Estimation• Hypothesis Testing

– Comparison– Association– Multivariable data analysis

• By Level of Underlying Distribution– Parametric Statistics– Non-parametric Statistics

Sampling Techniques

Generalization/Inferential Statistics

19

Descriptive Statistics

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• Measure of Location (Categorical Vars)– Frequency ( f )

• Measure of Location (Continuous Vars)– Mean

– Median

– Mode

Average

Mid-point

The Most Frequent

X1 X2 X3 X4 X5 X6 X7 X8 X9

xx

n

ii

n

= =∑

1 or μ = =∑

x

N

ii

n

1

X1 X2 X3 X4 X5 X6 X7 X8 X9

( 1 2 2 2 2 3 3 4 5)

Gender

MaleFemale

Coun

t

270

260

250

240

230

220

210

Male

Female

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Descriptive Statistics• Measure of Spread

– Range

– Standard Deviation / Variance

Max - Min

Xis deviate from Mean

xσ

μ=

−=

∑ ( )x

N

ii

n

1

Sx

n

ii

n

X=

−=

∑ ( )

- 1 1

or

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• Measure of Shape– Normal Distribution

– Skewed Distribution• Positively skewed Negatively skewed

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1. Skew = 0; “symmetric”;

median = mean

2. Skew > 0; “positive” or “right”skewed;

median<mean

3. Skew < 0; “negative” or “left”skewed;

median>mean


24

Example of Descriptive Statistics

25


Mon

thly

acc

iden

tal d

eath

s (U

S)

time0 20 40 60 80

6000

8000

10000

12000

Monthly

accid

ental deaths (U

time0 20 40 60 80

6000

8000

10000

12000

Month

ly a

ccid

enta

l death

s (

U

time0 20 40 60 80

6000

8000

10000

12000

Presentation matters!

1

7

12

1 1

7 7

12

12

26


Information can be seen in two dimensions that isn’t obvious in one dimension

Fra

ctio

n

No. of eggs10000 12000 14000

0

.05

.1

.15

Fra

ctio

n

weight4 6 8 10

0

.1

.2

.3N

o. o

f e

gg

s

weight4 6 8 10

10000

12000

14000

27

Inferential Statistics

28

Inferential Statistics

Ho: X1 = X2 Ho: μ1 = μ2

Ho: rxy = 0 Ho: ρ xy = 0

X μproportion Π

• Purpose of Inferential Statistics– Generalisabiliy of research results from Sample Statistics

to Population Parameters

• Types of Inferential Statistical Methods– Parameter Estimation - to estimate the range of values that is likely

to include the true value in population

– Hypothesis Testing - to ask whether an effect (difference) is present or not among different groups

29

Statistical Methods:Estimation

30

Parameter EstimationPoint Estimates:Single values (Mean, Variance, Correlation, treatment effect, relative risk, etc.) representing characteristics in the whole population

Interval Estimates: Ranges of values, usually centered around point estimates, indicating bounds within which we expect the true values for the whole population to lie (stability of the estimate)

Examples:• US adults have used unconventional therapy = 34%(31% - 37%, 95%CI)

•Sensitivity of clinical examination of splenomegaly =27% (19 - 36%, 95%CI)

• Relative risk of lung cancer of smoker vs. non-smoker = 5.6(2.1 - 8.9, 95%CI)

• Relative risk of HIV infected of male vs. female = 2.1 (0.5 - 6.9,95%CI)

31

Parameter EstimationApplication of Point Estimates & Confidence Intervals:

• Point estimates and confidence intervals are used to characterise the statistical precision of any rates (incidence, prevalence), comparisons of rates (e.g., relative risk), and other statistics.

• The narrower the CI, the more certain one can be about the size of the true effect. The true value is most likely to be close to the point estimate, less likely to be near the outer limits of the interval, and could (5 times out of 100) fall outside these limits altogether.

• CIs allow the reader to see the range of plausible values to decidewhether an effect size they regard as clinically meaningful is consistent with or rule out by the data.

32

Parameter Estimation

• Consideration in confidence level of the estimate

Confidence Interval, usually set at 95% CI, can be interpreted such that - if the study is unbiased and repeated 100 times, there is 95% chance that the true value is included in these intervals

Select a cut-point for CI

33

Example of

Estimate&

95%CI

34

Example of

Estimate&

95%CI

35

Statistical Methods:Hypothesis Testing

• Comparison among different groups

• Association among variables

• Multi-variable Data Analysis (MDA)(Statistical Modeling)

36

Statistical Methods:Hypothesis Testing - Comparisons

37

Hypothesis Testing Comparison of Qualitative Outcome Variable

Total Sample (N)

No exposure Group (N2)

Exposure Group (N1)

No Outcome

(n12)

No Outcome

(n22)

Outcome (n11)

Outcome (n21)

38

·

Outcome No Outcome

Exposure

No Exposure


(n11) (n12)

(n21) (n22)

(N1)

(N2)

(N)

Cross-sectional

39

·

Outcome No Outcome

Exposure

No Exposure


(n11) (n12)

(n21) (n22)

(N1)

(N2)

(N)

Cohort

40

·

Outcome No Outcome

Exposure

No Exposure


(n11) (n12)

(n21) (n22)

(N1)

(N2)

(N)

Case-Control

41

Hypothesis Testing Comparison of Quantitative Outcome Variable

Total Sample (N)

No exposure Group (N2)

Exposure Group (N1)

Continuous Outcome

(N1)

ContinousOutcome

(N2)

42

Hypothesis Testing Comparison of Quantitative Variable (Outcome)

•• Scatter Plot of the dataScatter Plot of the data

• Do you observe a similar amount of variation within each sample?• What about the weight gains? Is there any evident difference?

43


••SideSide--byby--side Box plotsside Box plots

• The dots represent the sample average in each group of observations.• Does this graph show any difference among the groups?

44


••SideSide--byby--side Box plotsside Box plots

The large variation within groups in the top figure suggests that the difference among the sample medians could be simply due to chance variability.

The data in the picture below are much more convincing that the populations differ.

( )21 −=

xxt

S12/n1 + S2

2/n2

45

Hypothesis Testing - Comparison

• Basic steps in hypothesis testing

46

Hypothesis Testing - Comparisonat α = 0.05Reject H0 !!

μ1 > μ2

H0: μ1 − μ2 = 0 Ha: μ1 − μ2 = 0

given n = very large

47


• Considerations in hypothesis testing1. What is your outcome of interest?2. What/who are the comparison groups?4. What is/are the hypothesis/hypotheses to be

tested?3. What is a cut-point for decision making about

the hypothesis/hypotheses?

48


• Hypothesis & Tail of the test– One-sided vs. Two-sided Test

Two-sided test:Ex Ho: Outcome 1 = Outcome 2

Ha: Outcome 1 ≠ Outcome 2One-sided test:

Ex Ho: Outcome 1 ≤ Outcome 2Ha: Outcome 1 > Outcome 2Ho: Outcome 1 ≥ Outcome 2Ha: Outcome 1 < Outcome 2

O1<O2 | O1=O2 | O1>O22.5% 95% 2.5%

O1<O2 | O1 >= O25% 95%

49


• Cut-point for decision making (p-value)The p-value is a quantitative estimate of the probability that observed differences in treatment effects in the particular study at hand could have happened by chance alone, assuming that there is in fact no difference between the groups.Or,If there is no difference between treatments and the trial was repeated many times, what proportion of the trials would lead to the conclusion that a treatment is as or more effective than found in the study?

Customary convention: p-value is set at 0.05• < 1 chance in 20 is a small risk of being wrong in decision making process• At the rate of 1/20, it is reasonable to conclude that such an occurrence is unlikely to arisen by chance alone

• Some researchers prefer to report the exact probabilities (e.g., 0.03, 0.001,0.11), rather than lumping them into two categories, p < 0.05 or p > 0.05.

50

Hypothesis Testing - Comparisonat α = 0.05Reject H0 !!

μ1 > μ2

H0: μ1 − μ2 = 0 Ha: μ1 − μ2 = 0

α / 2 = 0.005

− 2.576

α / 2 = 0.005

2.576

at α = 0.01Not Reject H0 !!

μ1 = μ2

given n = very large

p-value = 0.04

51

Exa

mpl

e of

Com

pari

sons

52

Example ofComparisons

53

Preliminary Results of the Phase III Efficacy Trial of AIDSVAX B/B February 23, 2003

All Volunteers 98/1679 (5.8%) 191/3330 (5.7%) 3.8% (-22.9 to 24.7%)

White & Hispanic 81/1508 (5.4%) 179/3003 (6.0%) -9.7% (-42.8 to 15.7%)

Black/Asian/Other 17/171 (9.9%) 12/327 (3.7%) 66.8% (30.2 to 84.2%)*66.8% (30.2 to 84.2%)*

Black 9/111 (8.1%) 4/203 (2.0%) 78.3% (29.0 to 93.3%)**78.3% (29.0 to 93.3%)**

Asian 2/20 (10.0%) 2/53 (3.8%) 68.0% (-129.4 to 95.5%)

Other 6/40 (15.0%) 6/71 (8.5%) 46.2% (-67.8 to 82.8%)

Vaccine EfficacyWeighted cohort Placebo Inf./total Vaccine Inf./total VE (95.12%CI)

** p <0.02* p <0.01

Example of Comparisons

54

Example ofComparisons

∗∗∗

55

Statistical Methods:Hypothesis Testing - Associations

56

Associations

Statistics are also used to describe the degree of association between variables.• Pearson’s product moment correlation for interval data• Spearman’s rank correlation for ordinal data• Odds Ratio , Relative Risk for ordinal or nominal data • etc.

These statistics expressed in quantitative terms the extent to which the value of one variable is associated with the value of the other.

The association coefficient has a corresponding statistical test to assess whether the observed association is greater than might have arisen by chance alone.

57

AssociationsQuantitative Variables

Thig

h c

ircum

fere

nce (

cm

Knee circumference (cm)30 35 40 45 50

40

60

80

100

Many interesting features can be seen in a scatterplot including the overall pattern (i.e. linear, nonlinear, periodic), strength and direction of the relationship, and outliers (values which are far from the bulk of the data).

58

Associations Quantitative Variables

y

x-2 -1 0 1 2

-2

-1

0

1

2

y

x-2 -1 0 1 2

-2

-1

0

1

2

y

x-2 -1 0 1 2

0

1

2

3

4

Perfect positive correlation

(R = 1)

Perfect negative correlation

(R = -1)

Uncorrelated (R = 0)

but dependent

59


y

x0 2 4 6

.4

.6

.8

1

1.2

Scatterplot showing nonlinear relationship

s1

s30 20 40 60

0

20

40

60

O

Scatterplot showing daily rainfall amount (mm) at nearby stations in SW Australia.

Note outliers -- Are they data errors … or interesting science?!

60

Scatterplot of bodyfat vs. height (inches)

HEIGHT

80706050403020

BO

DY

FAT

40

30

20

10

0


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Scatterplot of bodyfat vs. height (inches)

HEIGHT

787674727068666462

BO

DY

FAT

40

30

20

10

0


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1009080706050403020100

1009080706050403020

10

Pla

que

inde

x

Brushing quality

Scatterplot showing

linear vs. nonlinear relationship

63


Scatterplot showing

linear vs. nonlinear relationship

1009080706050403020100

100908070605040302010

Pla

que

inde

x

Brushing quality

{ { { {{1 2 3 4 5

64

Example ofAssociations

65

Example ofAssociations

66

Associations Qualitative Variables

To assess whether two factors are related, construct an R x C table that cross-classifies the observations according to the two factors.

Example. Education versus willingness to participate in a study of a vaccine to prevent HIV infection if the study was to start tomorrow. Counts, percents and row and column totals are given.

d e f i n i t e l yn o t

p r o b a b l yn o t

P r o b a b l y d e f i n i t e l y T o t a l

< h i g hs c h o o l

5 21 .1 %

7 91 .6 %

3 4 27 .0 %

2 2 64 .6 %

6 9 9

h i g h s c h o o l 6 21 .3 %

1 5 33 .2 %

4 1 78 .6 %

2 6 25 .4 %

8 9 4

s o m ec o l l e g e

5 31 .1 %

2 1 34 .4 %

6 2 91 3 .0 %

3 7 57 .7 %

1 2 7 0

c o l l e g e 5 41 .1 %

2 3 14 .8 %

5 7 11 1 .8 %

2 4 45 .0 %

1 1 0 0

s o m e p o s tc o l l e g e

1 80 .4 %

4 60 .9 %

1 3 92 .9 %

7 41 .5 %

2 7 7

g r a d u a t e /p r o f

2 50 .5 %

1 3 92 .9 %

3 3 06 .8 %

1 1 62 .4 %

6 1 0

T o t a l 2 6 4 8 6 1 2 4 2 8 1 2 9 7 4 8 5 0

The table displays the joint distribution of education and willingness to participate.

67

Associations Qualitative Variables

The marginal distributions of a two-way table are simply the distributions of each measure summed over the other.

E.g. Willingness to participateDefinitelynot

Probablynot

Probably Definitely

264 861 2428 12975.4% 17.8% 50.1% 26.7%

A conditional distribution is the distribution of one measure conditional on (given the) value of the other measure.

E.g. Willingness to participate among those with a college education.

Definitelynot

Probablynot

Probably Definitely

54 231 571 2444.9% 21.0% 51.9% 22.2%

68

Example of Associations

69


Odds = P (disease) / P (no disease)= (disease/total) / (no disease/total)

Odds Ratio = Odds Grp1 / Odds Grp2

Odds (<=18,500 cpm) = 7.5|92.5 = .08Odds (> 18,500 cpm) = 29.6|70.4 = .42Odds Ratio = .42/.08 = 5.25

(8/107) / (99/107)

70

Example of

Associations

71


Risk = Incidence of disease= new cases / total exposures

RR = Risk Grp1 / Risk Grp2

Risk (never) = 7/47 = .149Risk (sporadic) = 9/35 = .257RR (sporadic vs.never) = 25.7/14.9 = 1.72

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73

Statistical Methods:Multi-variable Data Analysis

74

Multi-variable Data Analysis Types of Epidemiological/Clinical Studies

75

Multi-variable Data Analysis Causes/Exposures vs. Outcomes

76


crowding

Malnutrition

Vaccination

Genetic

Risk factors for tuberculosis(Distant from Outcome)

Mechanism of Tuberculosis(Proximal to Outcome)

Susceptible Host

Susceptible Host

InfectionInfection TuberculosisTuberculosisTissue Invasion

and Reaction

Exposure to Mycobacterium

ExampleExample:: Causes of TuberculosisCauses of Tuberculosis

77


Example: Relationship between risk factors and disease : hypertension ( BP) and congestive heart failure (CHF). Hypertension causes many diseases, including congestive heart failure, and congestive heart failure has many causes, including hypertension.

78

Multi-variable Data AnalysisStatistical Modelling

Most clinical outcomes are the result of many variables acting together in complex ways.

Basic steps in performing MDA:• First try to understand the relationships by examining relatively simple arrangement of the data (e.g., 2 x 2 table), examining whether the effect of one variable is changed by the presence or absence of one or more other variables (stratified analysis).

• In addition to contingency tables, examine the effects of several variables simultaneously at one time using “multi-variable modelling” methods.

79


• Considerations in MDA• What are the right variables to be included in the model?• What are the right amount of variables to be included in the model?• What is the best model to explain the phenomenon?

80


• the right variables to be included in the model• identify and measure all the variables that might

be related to the outcome of interest• What is the definition of each variable?• How do you make decision in classifying your subjects?

• the right amount of variables• reduce the number of variables to be considered in the model to manageable number, usually no more than several.

• Often this is done by selecting variables that are, when taken one at a time, most strongly related to the outcome. • Evidence for the biological importance of the variable is alsoconsidered in making the selection.

81


• the “best” model• try different models and select the theoretical model that is best fitted to the data

• Statistical model may vary from one data-set to another. The validity of the model is based on the assumptions about the data that may not be met or it is sensitive to random variation. • The model is thus needed to be validated. This is commonly done by seeing if the model predicts what is found in another independent sample of patients. The results of the first model are considered a hypothesis, to be tested by new data.

82

Exampleof

MDA

83

Example of MDA

84

Exampleof

MDA

85

Example of MDA

86

Example of MDA

87

IIIChoosing the Right Statistical Procedure

88

Choosing the Right Data Analysis Procedure

Basic questions that you should have answers before you can choose the correct test are the following:

What is the purpose of the analysis?- describe the data; Or- compare groups of data to make decisions; or- examine the association between variables for

prediction or forecasting?

Is the distribution of the data approximately normal?

Is the sample size large enough that the Central Limit Theoremwill allow a normality assumption?

89

Selecting a Statistical Method

Goal Type of Outcome Dataof the Continuous Categorical Binomial Survivalanalysis (from Gaussian Continuous Time

Population) (Non-Gaussian)

DescribeValue of Data

(1 Group)

CompareValue of Data

vs. Hypothetical

Value(1 Group)

Mean,SD

Median,Interquar-tile range

Proportion(Percent)

Kaplan-Meier survival curve

One-sample t-test

Wilcoxon’s test

Chi-square (χ2) orBinomial/ Runs test

90




CompareValues 2 Grpsof Indept >2 GrpsGrps.

CompareValues 2 Grpsof Paired >2 GrpsGrps/Vars.

Unpaired t-test

One-way ANOVA

Mann-Whitney test

Kruskal-Wallis test

χ2 test , Fisher’s Exact,

χ2 test

Log-rank / Mantel-HaenszelCox Prop Haz.Reg.

Paired t-test

Repeated measures ANOVA

Wilcoxon’s test

Friedman’s test

McNemar’s χ2

test

Cochrane’s Q test

Condtn Prop Haz.Reg.

Condtn Prop Haz.Reg.

91




Quantify Association

Values of Two variables(1 Group)

Predict Value of Outcome Var:from 1 Var

(Simple Reg. )from > 2 Vars(Multiple Reg.)

Pearson’s Correlation

Spearman’s Correlation

Contingency coefficient, Crude Odds Ratio, Relatv Risk

Linear orNon-linear

Regression.

Cox’s Proportional Haz. Reg.

Non-parametricRegression

Logistic Regression

92




Measures of Agreement Values

from Two Raters/Methods

Measures of Validity

Values from Two Raters/Methods

Pearson’s Correlation

Weighted Kappa (κ)

Weighted Kappa (κ)

Agreement rate

Cohen’s κ

ICC

ANOVA

Factor Analysis

Non-parametric ANOVA

Factor Analysis

χ2

SensitivitySpecificityROC curve

93

Example: Selecting a Statistical Method

94

Exa

mpl

e: S

elec

ting

a St

atis

tical

Met

hod

r = 0.31

r = 0.05

95

Exa

mpl

e: S

elec

ting

a St

atis

tical

Met

hod

96


Dose-Response

97


Charactersitcs No. of No. of Patients 7-Year survival, Rate ratio patients who died % % (95% CI)* (95% CI)** Age at Infection, years <= 19 105 31 (29.5) 72.3 (62.1-80.1) Referent >=20 89 35 (39.3) 63.3 (50.5-73.6) 1.50 (0.92-2.45) Sex work Brothel 159 54 (34.0) 69.6 (61.1-76.6) 1.34 (0.71-2.52) Nonbrothel 35 12 (34.3) 62.9 (41.6-78.2) Referent Oral contraceptive use Yes 112 36 (32.1) 69.6 (59.2-77.9) 0.83 (0.51-1.36) No 82 30 (36.6) 67.1 (54.7-76.8) Referent Depot medroxyprogesterone use Yes 55 18 (32.7) 70.9 (55.7-81.7) 0.78 (0.45-1.37) No 139 48 (34.5) 67.8 (58.4-75.5) Referent Infection status Seroconverted 34 7 (20.6) *** 1.42 (0.63-3.22) Seropositive at enrl. 160 59 (36.9) 69.6 (61.4-76.4) Referent Viral load, HIV-1 RNA copies/ml. >1000,000 34 24 (70.6) 34.5 (18.8-50.9) 15.40 (5.2-45.2) 10,000-100,000 113 38 (33.6) 70.3 (60.1-78.4) 4.63 (1.64-13.1) <10,000 47 4 (8.5) 92.5 (78.4 -97.5) Referent Total 194 66 (34.0) 68.7 (61.0-75.2) * Survival analysis ** Cox proportional hazard model *** Insufficient follow-up time to this m ore recent converted group; 5-year survival = 77.8 (56.8-89.5) %

Table 2. Survival from time of infection of 194 HIV-infected CSWs

98

Example: Selecting a Statistical MethodSu

rviv

al (%

)

Months

10 20 30 40 50 60 70 80 90 100 110 1200

20

40

60

80

100

A B

Months

10 20 30 40 50 60 70 80 90 100 110 120

20

40

60

80

100 <10,000

>100,000

10,000-100,000

00 0

Surv

ival

(%)

Figure 2. Survival from time of human immunodeficiency virus (HIV) infection of 194 CSWs.

A, Overall. B, By serum virus load (HIV type 1 RNA copies/mL).

Each curve is truncated when <10 women remain in that group.

99


Charactersitcs No. of No. of Pts 5-Year survival, Rate ratio Adjusted rate ratiopatients who died % % (95% CI)* (95% CI)** (95% CI)***

Initial CD4 lymphocyte, cells/μL<200 15 14 (93.3) 0 20.9 (9.00-48.7) 15.5 (6.46-37.4)200-500 88 34 (38.6) 63.4 (51.8-72.9) 2.46 (1.21-5.01) 1.42 (0.67-3.00)>500 54 10 (18.5) 84.7 (70.4-92.4) Referent Referent

Viral load, HIV-1 RNA copies/ml.>1000,000 30 23 (76.7) 26.7 (12.6-43.0) 13.9 (4.78-40.6) 12.5 (4.09-38.2)10,000-100,000 89 31 (34.8) 65.0 (53.1-74.5) 3.87 (1.36-11.0) 3.42 (1.19-9.81)<10,000 38 4 (10.5) 96.7 (78.6-99.5) Referent Referent

Total 157 58 (36.9) 64.6 (56.0-71.9)

* Survival analysis** Cox proportional hazard model*** Cox proporational hazard model adjusted for initial CD4 lymphocyte count and virus load

Table 3. Mortality from time of first CD4 T lymphocyte count of 157 HIV-infected CSWs (125 women were HIV seropositive at study enrollment and 32 seroconvertedduring study).

overview of inferential statistics - interfetpthailand chavez/inferential... · 2 topics • bias...

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