screen - Christophe Lalanne

Découverte du logiciel Stata
Mesures et tests d’association
Christophe Lalanne
www.aliquote.org
Synopsis
Tests de comparaison de deux moyennes
Tests de comparaison de k moyennes
Tests de comparaison de deux proportions
Analyse d’un tableau de contingence
Mesures d’association en épidémiologie
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Données d’illustration
Enquête socio-économique allemande réalisée en 2009 :
« GSOEP » (3) .
ybirth
hhnr2009
sex
mar
edu
yedu
voc
emp
egp
income
hhinc
size
hhsize
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Données socio-démographiques
année de naissance
foyer résidentiel
sexe
statut marital
niveau d’éducation
nombre d’années de formation
niveau secondaire ou université
Emploi et revenu
type d’emploi
catégorie socio professionnelle
revenus (€)
revenus du foyer (€)
Logement
taille du logement
nombre de personnes dans habitation
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Fichier de données : gsoep09.dta
. use data/gsoep09
(SOEP 2009 (Kohler/Kreuter))
Pré-traitements :
. gen age = 2009 - ybirth
. mvdecode income, mv(0=.c)
income: 1369 missing values generated
. gen lincome = log(income)
(2001 missing values generated)
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Tests de comparaison de deux moyennes
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Comparaison de deux moyennes
Le test de Student, via la commande ttest, s’utilise dans le cas
des comparaisons de moyennes pour un échantillon (H0 : µ =
0) ou deux échantillons (indépendants ou non).
Illustration : le revenu moyen diffère-t-il selon le sexe ?
. bysort sex: summarize lincome
-------------------------------------------------------------------------------> sex = Male
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------lincome |
1746
10.08129
1.083648
3.828641
13.70765
-------------------------------------------------------------------------------> sex = Female
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------lincome |
1664
9.443893
1.073004
5.09375
13.32572
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. graph box lincome, over(sex) ytitle("Income (log(2)")
14
Income (log(2)
12
10
8
6
4
Male
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Female
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Test de Student
Statistics . Summaries, tables, and tests . Classical tests of hypotheses . t test
. ttest lincome, by(sex)
Two-sample t test with equal variances
-----------------------------------------------------------------------------Group |
Obs
Mean
Std. Err.
Std. Dev. [95% Conf. Interval]
---------+-------------------------------------------------------------------Male |
1746
10.08129
.0259338
1.083648
10.03043
10.13216
Female |
1664
9.443893
.0263042
1.073004
9.3923
9.495486
---------+-------------------------------------------------------------------combined |
3410
9.770257
.0192551
1.124407
9.732504
9.808009
---------+-------------------------------------------------------------------diff |
.6374003
.0369475
.5649587
.7098419
-----------------------------------------------------------------------------diff = mean(Male) - mean(Female)
t = 17.2515
Ho: diff = 0
degrees of freedom =
3408
Ha: diff < 0
Pr(T < t) = 1.0000
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Ha: diff != 0
Pr(|T| > |t|) = 0.0000
Ha: diff > 0
Pr(T > t) = 0.0000
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Test de Student (bis)
Sans supposer l’égalité des variances parentes (correction de
Satterthwaite, option unequal) (5) :
. ttest lincome, by(sex) welch
Two-sample t test with unequal variances
-----------------------------------------------------------------------------Group |
Obs
Mean
Std. Err.
Std. Dev. [95% Conf. Interval]
---------+-------------------------------------------------------------------Male |
1746
10.08129
.0259338
1.083648
10.03043
10.13216
Female |
1664
9.443893
.0263042
1.073004
9.3923
9.495486
---------+-------------------------------------------------------------------combined |
3410
9.770257
.0192551
1.124407
9.732504
9.808009
---------+-------------------------------------------------------------------diff |
.6374003
.0369388
.5649759
.7098247
-----------------------------------------------------------------------------diff = mean(Male) - mean(Female)
t = 17.2556
Ho: diff = 0
Welch's degrees of freedom = 3405.02
Ha: diff < 0
Pr(T < t) = 1.0000
Ha: diff != 0
Pr(|T| > |t|) = 0.0000
Ha: diff > 0
Pr(T > t) = 0.0000
Si l’on souhaite vraiment comparer deux variances, la commande sdtest offre la même syntaxe que ttest.
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Intervalles de confiance
La commande ci permet de construire des intervalles de fluctuation pour un certain niveau de confiance (level()) :
. bysort sex: ci lincome
-------------------------------------------------------------------------------> sex = Male
Variable |
Obs
Mean
Std. Err.
[95% Conf. Interval]
-------------+--------------------------------------------------------------lincome |
1746
10.08129
.0259338
10.03043
10.13216
-------------------------------------------------------------------------------> sex = Female
Variable |
Obs
Mean
Std. Err.
[95% Conf. Interval]
-------------+--------------------------------------------------------------lincome |
1664
9.443893
.0263042
9.3923
9.495486
Commande additionnelle : mean (idem, utilisation de la loi normale pour les IC à 95 %).
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. mean lincome if sex == 1
Mean estimation
Number of obs
=
1746
-------------------------------------------------------------|
Mean
Std. Err.
[95% Conf. Interval]
-------------+-----------------------------------------------lincome |
10.08129 .0259338
10.03043
10.13216
--------------------------------------------------------------
Manuellement :
. local zc = 1-invnormal(0.95)
. display 10.08129 - `zc'/2 * .0259338
10.089652
Si l’on souhaite construire des intervalles de confiance basés
sur une distribution de Student, on utilisera plutôt invt (tprob
fournit les valeurs de probabilités au lieu des fractiles) :
. display 10.08129 - invt(1745, 0.975) * .0259338
10.030425
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Alternative non-paramétrique
Le test de Wilcoxon (différent de median) constitue une alternative non-paramétrique au test de Student.
. ranksum lincome, by(sex)
Two-sample Wilcoxon rank-sum (Mann-Whitney) test
sex |
obs
rank sum
expected
-------------+--------------------------------Male |
1746
3551869.5
2977803
Female |
1664
2263885.5
2837952
-------------+--------------------------------combined |
3410
5815755
5815755
unadjusted variance
adjustment for ties
adjusted variance
8.258e+08
-16.745225
---------8.258e+08
Ho: lincome(sex==Male) = lincome(sex==Female)
z = 19.976
Prob > |z| =
0.0000
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Tests de comparaison de k moyennes
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Analyse de variance à un facteur
L’analyse de variance (ANOVA) est utilisée pour comparer plus
de 2 moyennes (H0 : µ1 = µ2 = · · · = µk ). Stata offre deux
commandes (sans passer par le modèle linéaire) : oneway et
anova.
Illustration : le revenu moyen diffère-t-il selon le type d’emploi ?
. recode egp (1/2=1) (3/5=2) (8/9=3) (15/18=.) , ///
gen ( egp4 )
. label define egp4 1 " Service class 1/2" ///
2 " Non - manuals & self - employed " 3 " Manuals "
. label values egp4 egp4
(4435 differences between egp and egp4)
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Distributions par groupe
. histogram lincome, by(egp4, col(3)) freq
Service class 1/2
Non−manuals & self−employed
Manuals
200
Frequency
150
100
50
0
5
10
15
5
10
15
5
10
15
lincome
Graphs by RECODE of egp (Social Class (EGP))
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. twoway (kdensity lincome), by(egp4)
Service class 1/2
Non−manuals & self−employed
.8
.6
kdensity lincome
.4
.2
0
5
10
15
Manuals
.8
.6
.4
.2
0
5
10
15
x
Graphs by RECODE of egp (Social Class (EGP))
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. graph box lincome, over(egp4) ytitle("Income (log(2)")
14
Income (log(2)
12
10
8
6
4
Service class 1/2
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Non−manuals & self−employed
Manuals
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Moyennes conditionnelles
. tabstat lincome, by(egp4) stats(mean sd count)
Summary for variables: lincome
by categories of: egp4 (RECODE of egp (Social Class (EGP)))
egp4 |
mean
sd
N
-----------------+-----------------------------Service class 1/ | 10.29525 .9454878
1085
Non-manuals & se | 9.776857 .9735212
868
Manuals | 9.615197 1.002863
1102
-----------------+-----------------------------Total | 9.902652 1.018826
3055
------------------------------------------------
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Tableau d’ANOVA
Statistics . Linear models and related . ANOVA/MANOVA . Oneway ANOVA
. oneway lincome egp4
Analysis of Variance
Source
SS
df
MS
F
Prob > F
-----------------------------------------------------------------------Between groups
272.026782
2
136.013391
143.24
0.0000
Within groups
2898.0461
3052
.949556388
-----------------------------------------------------------------------Total
3170.07288
3054
1.03800684
Bartlett's test for equal variances:
chi2(2) =
3.7888
Prob>chi2 = 0.150
oneway [response_var] [factor_var] [if] [in] [ , options]
• tabulate : affichage des moyennes, écarts-type et
effectifs
• bonferroni : comparaison des paires de moyennes avec
correction de Bonferroni
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Vérification des conditions d’application
• indépendance des observations
• normalité des résidus
• égalité des variances (parentes)
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Normalité des résidus
La commande swilk fournit le test de Shapiro-Wilks. Mais en
règle générale, les méthodes graphiques sont préférables :
. quietly: anova lincome egp4
. predict r, resid
. qnorm r
(2356 missing values generated)
4
Residuals
2
0
−2
−4
−6
−4
−2
0
2
4
Inverse Normal
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Égalité des variances
Stata fournit le résultat du test de Bartlett pour l’égalité des
variances avec la commande oneway. Le test de Levenne s’obtient avec la commande robvar (W0) :
. robvar lincome, by(egp4)
RECODE of |
egp (Social |
Class |
Summary of lincome
(EGP)) |
Mean
Std. Dev.
Freq.
------------+-----------------------------------Service c |
10.295247
.94548776
1085
Non-manua |
9.7768571
.97352115
868
Manuals |
9.6151967
1.0028632
1102
------------+-----------------------------------Total |
9.9026521
1.0188262
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W0
= 12.5051486
df(2, 3052)
Pr > F = 0.00000390
7.9388574
df(2, 3052)
Pr > F = 0.00036403
W10 = 10.6968625
df(2, 3052)
Pr > F = 0.00002348
W50 =
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Comparaison de paires de moyennes
Option de correction pour les tests post-hoc : bonferroni, scheffe
ou sidak.
. oneway lincome egp4, bonferroni noanova
Comparison of lincome by RECODE of egp (Social Class (EGP))
(Bonferroni)
Row Mean-|
Col Mean |
Service
Non-manu
---------+---------------------Non-manu |
-.51839
|
0.000
|
Manuals |
-.680051
-.16166
|
0.000
0.001
On arrive à des conclusions similaires en appliquant la correction de Bonferroni sur les résultats de simples tests de Student.
. quietly: ttest lincome if egp4 != 1, by(egp4)
. display r(p)*3
.0009856
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Alternative à oneway
La commande oneway est limité au cas à un facteur explicatif.
La commande anova est plus générale et couvre : les plans factoriels et emboîtés, les plans équilibrés ou non (cf. calcul des
sommes de carrés), les mesures répétées, l’analyse de covariance.
. anova lincome egp4
Number of obs =
3055
Root MSE
= .974452
R-squared
=
Adj R-squared =
0.0858
0.0852
Source | Partial SS
df
MS
F
Prob > F
-----------+---------------------------------------------------Model | 272.026782
2 136.013391
143.24
0.0000
|
egp4 | 272.026782
2 136.013391
143.24
0.0000
|
Residual |
2898.0461 3052 .949556388
-----------+---------------------------------------------------Total | 3170.07288 3054 1.03800684
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Comparaisons multiples
En utilisant anova, les comparaisons par paires de moyennes
s’obtiennent à l’aide de pwcompare, commande plus générale
que pwmean. Les options de correction (mcompare()) incluent en
plus : tukey, snk, duncan et dunnett.
. pwcompare egp4, cformat(%3.2f)
Pairwise comparisons of marginal linear predictions
Margins
: asbalanced
-----------------------------------------------------------------------------|
Unadjusted
|
Contrast
Std. Err.
[95% Conf. Interval]
-----------------------------+-----------------------------------------------egp4 |
Non-manuals & self-employed |
vs |
Service class 1/2 |
-0.52
0.04
-0.61
-0.43
Manuals |
vs |
Service class 1/2 |
-0.68
0.04
-0.76
-0.60
Manuals |
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vs |
Tests de comparaison de deux proportions
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Tests de proportion exact et approché
Outre le test du χ2 de Pearson dans le cas du croisement de
deux variables binaires, Stata dispose des commandes bitest
(test binomial) et prtest (test reposant sur l’approximation normale). Dans le cas univarié, la variable binaire doit être codée
en 0/1. Plusieurs types d’intervalles de confiance sont disponibles (4) .
Illustration : distribution équilibrée des deux sexes dans l’échantillon.
. generate sexb = sex - 1
. tabulate sexb
sexb |
Freq.
Percent
Cum.
------------+----------------------------------0 |
2,585
47.77
47.77
1 |
2,826
52.23
100.00
------------+----------------------------------Total |
5,411
100.00
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Test binomial
Statistics . Summaries, tables, and tests . Classical tests of hypotheses . Proportion test
. bitest sexb == 0.5
Variable |
N
Observed k
Expected k
Assumed p
Observed p
-------------+-----------------------------------------------------------sexb |
5411
2826
2705.5
0.50000
0.52227
Pr(k >= 2826)
= 0.000551
Pr(k <= 2826)
= 0.999500
Pr(k <= 2585 or k >= 2826) = 0.001102
(one-sided test)
(one-sided test)
(two-sided test)
. ci sexb, binomial
-- Binomial Exact -Variable |
Obs
Mean
Std. Err.
[95% Conf. Interval]
-------------+--------------------------------------------------------------sexb |
5411
.5222695
.0067905
.508859
.5356559
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Test de proportion pour un échantillon
Statistics . Summaries, tables, and tests . Classical tests of hypotheses . Binomial probability test
. prtest sexb == 0.5
One-sample test of proportion
sexb: Number of obs =
5411
-----------------------------------------------------------------------------Variable |
Mean
Std. Err.
[95% Conf. Interval]
-------------+---------------------------------------------------------------sexb |
.5222695 .0067905
.5089604
.5355785
-----------------------------------------------------------------------------p = proportion(sexb)
z =
3.2763
Ho: p = 0.5
Ha: p < 0.5
Pr(Z < z) = 0.9995
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Ha: p != 0.5
Pr(|Z| > |z|) = 0.0011
Ha: p > 0.5
Pr(Z > z) = 0.0005
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Test de proportion pour deux échantillons
. generate egpb = egp4 == 1
. prtest egpb, by(sexb)
Two-sample test of proportions
0: Number of obs =
2585
1: Number of obs =
2826
-----------------------------------------------------------------------------Variable |
Mean
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------0 |
.2201161 .0081491
.2041441
.236088
1 |
.1854211 .0073107
.1710923
.1997498
-------------+---------------------------------------------------------------diff |
.034695
.0109478
.0132376
.0561523
| under Ho: .0109269
3.18
0.001
-----------------------------------------------------------------------------diff = prop(0) - prop(1)
z =
3.1752
Ho: diff = 0
Ha: diff < 0
Pr(Z < z) = 0.9993
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Ha: diff != 0
Pr(|Z| > |z|) = 0.0015
Ha: diff > 0
Pr(Z > z) = 0.0007
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Commandes immédiates
Plusieurs commandes Stata acceptent des formes « immédiates ».
prtesti # obs1 # p1 # obs2 # p2 [ , levels (#) count ]
Statistics . Summaries, tables, and tests . Classical tests of hypotheses . Proportion test calculator
. prtesti 2585 0.2201 2826 0.1854
L’option count permet de travailler avec les effectifs observés
plutôt que des fréquences relatives.
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Analyse d’un tableau de contingence
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Construction d’un tableau 2x2
Statistics . Summaries, tables, and tests . Frequency tables .
Two-way table with measures of association
La commande tabulate (twoway) permet de construire un tableau d’effectifs ou de fréquences relatives et dispose d’options pour les statistiques de Pearson et de Fisher (1) .
. tabulate sex egp4
|
RECODE of egp (Social Class
|
(EGP))
Gender | Service c Non-manua
Manuals |
Total
---------------------+---------------------------------+---------Male |
569
290
717 |
1,576
Female |
524
592
396 |
1,512
---------------------+---------------------------------+---------Total |
1,093
882
1,113 |
3,088
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Profils ligne et colonne
. tabulate sex egp4, row
+----------------+
| Key
|
|----------------|
| frequency
|
| row percentage |
+----------------+
|
RECODE of egp (Social Class
|
(EGP))
Gender | Service c Non-manua
Manuals |
Total
---------------------+---------------------------------+---------Male |
569
290
717 |
1,576
|
36.10
18.40
45.49 |
100.00
---------------------+---------------------------------+---------Female |
524
592
396 |
1,512
|
34.66
39.15
26.19 |
100.00
---------------------+---------------------------------+---------Total |
1,093
882
1,113 |
3,088
|
35.40
28.56
36.04 |
100.00
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Test d’association du χ2
. tabulate sex egp4, chi
|
RECODE of egp (Social Class
|
(EGP))
Gender | Service c Non-manua
Manuals |
Total
---------------------+---------------------------------+---------Male |
569
290
717 |
1,576
Female |
524
592
396 |
1,512
---------------------+---------------------------------+---------Total |
1,093
882
1,113 |
3,088
Pearson chi2(2) = 196.5961
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Pr = 0.000
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Effectifs théoriques
L’option expected fournit les effectifs théoriques.
. tabulate sex egp4, expected
+--------------------+
| Key
|
|--------------------|
|
frequency
|
| expected frequency |
+--------------------+
|
RECODE of egp (Social Class
|
(EGP))
Gender | Service c Non-manua
Manuals |
Total
---------------------+---------------------------------+---------Male |
569
290
717 |
1,576
|
557.8
450.1
568.0 |
1,576.0
---------------------+---------------------------------+---------Female |
524
592
396 |
1,512
|
535.2
431.9
545.0 |
1,512.0
---------------------+---------------------------------+---------Total |
1,093
882
1,113 |
3,088
|
1,093.0
882.0
1,113.0 | 3,088.0
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Test exact de Fisher
. tabulate sex egp4, exact
Enumerating sample-space
stage 3: enumerations =
stage 2: enumerations =
stage 1: enumerations =
combinations:
1
351
0
|
RECODE of egp (Social Class
|
(EGP))
Gender | Service c Non-manua
Manuals |
Total
---------------------+---------------------------------+---------Male |
569
290
717 |
1,576
Female |
524
592
396 |
1,512
---------------------+---------------------------------+---------Total |
1,093
882
1,113 |
3,088
Fisher's exact =
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0.000
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Mesures d’association en épidémiologie
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Mesures de risque
Statistics . Epidemiology and related . Tables for epidemiologists
Stata offre une grande variété de tests d’association et de mesures de risque classiquement utilisées en épidémiologie.
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Odds-ratio
La commande tabodds s’utilise dans le cas des études castémoins ou des études transversales. Elle permet de calculer
l’odds-ratio et son intervalle de confiance asymptotique (autre
option : cornfield ou woolf), ainsi que tester l’homogénéité
des OR entre strates (test de Mantel-Haenszel).
Autres commandes disponibles : cc et mcc (étude cas-témoins),
ir (étude de cohorte). Toutes ces commandes disposent d’une
forme « immédiate » alternative.
Manuel : [ST] epitab
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Données d’illustration
Étude sur les poids de naisssance (2) .
low
age
lwt
race
smoke
ht
ui
ftv
ptl
bwt
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poids de naissance < 2,5 kg
âge de la mère
poids de la mère (livres) aux dernières règles
ethnicité de la mère (« w », « b », « o »)
statut fumeur de la mère pendant la grossesse
antécédent d’hypertension
présence d’irritabilité utérine
nb de visites chez le gynécologue 1er trimestre
nb d’accouchements pré terme antérieurs
poids du bébé (grammes)
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. clear all
. webuse lbw
(Hosmer & Lemeshow data)
. list in 1/5
1.
2.
3.
4.
5.
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+-----------------------------------------------------------------------+
| id
low
age
lwt
race
smoke
ptl
ht
ui
ftv
bwt |
|-----------------------------------------------------------------------|
| 85
0
19
182
black
nonsmoker
0
0
1
0
2523 |
| 86
0
33
155
other
nonsmoker
0
0
0
3
2551 |
| 87
0
20
105
white
smoker
0
0
0
1
2557 |
| 88
0
21
108
white
smoker
0
0
1
2
2594 |
| 89
0
18
107
white
smoker
0
0
1
0
2600 |
+-----------------------------------------------------------------------+
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Calcul de l’odds-ratio
. tabodds low smoke, or
--------------------------------------------------------------------------smoke | Odds Ratio
chi2
P>chi2
[95% Conf. Interval]
-------------+------------------------------------------------------------nonsmoker |
1.000000
.
.
.
.
smoker |
2.021944
4.90
0.0269
1.069897
3.821169
--------------------------------------------------------------------------Test of homogeneity (equal odds): chi2(1) =
4.90
Pr>chi2 =
0.0269
Score test for trend of odds:
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chi2(1) =
Pr>chi2 =
4.90
0.0269
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. cc low smoke, woolf
| smoked during pregnancy|
Proportion
|
Exposed
Unexposed |
Total
Exposed
-----------------+------------------------+-----------------------Cases |
30
29 |
59
0.5085
Controls |
44
86 |
130
0.3385
-----------------+------------------------+-----------------------Total |
74
115 |
189
0.3915
|
|
|
Point estimate
|
[95% Conf. Interval]
|------------------------+-----------------------Odds ratio |
2.021944
|
1.08066
3.783112 (Woolf)
Attr. frac. ex. |
.5054264
|
.0746392
.7356673 (Woolf)
Attr. frac. pop |
.2569965
|
+------------------------------------------------chi2(1) =
4.92 Pr>chi2 = 0.0265
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Calcul du risque relatif
. cs low smoke
| smoked during pregnancy|
|
Exposed
Unexposed |
Total
-----------------+------------------------+-----------Cases |
30
29 |
59
Noncases |
44
86 |
130
-----------------+------------------------+-----------Total |
74
115 |
189
|
|
Risk | .4054054
.2521739 |
.3121693
|
|
|
Point estimate
|
[95% Conf. Interval]
|------------------------+-----------------------Risk difference |
.1532315
|
.0160718
.2903912
Risk ratio |
1.607642
|
1.057812
2.443262
Attr. frac. ex. |
.377971
|
.0546528
.5907112
Attr. frac. pop |
.1921887
|
+------------------------------------------------chi2(1) =
4.92 Pr>chi2 = 0.0265
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Références I
1. I Campbell. Chi-squared and Fisher-Irwin tests of two-by-two tables with small
sample recommendations. Statistics in Medicine, 26(19) :3661–3675, 2007.
2. D Hosmer and S Lemeshow. Applied Logistic Regression. New York : Wiley, 1989.
3. U Kohler and F Kreuter. Data Analysis Using Stata. College Station : Stata Press,
2012.
4. RG Newcombe. Two-sided confidence intervals for the single proportion : comparison of seven methods. Statistics in Medicine, 17(8) :857–872, 1998.
5. BL Welch. On the comparison of several mean values : An alternative approach.
Biometrika, 38 :330–336, 1951.
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Index des commandes
anova, 24
bitest, 28
bysort, 6, 10
cc, 44
ci, 10, 28
clear, 42
cs, 45
display, 11, 23
generate, 4, 27, 30
graph box, 7, 17
histogram, 15
invt, 11
kdensity, 16
label define, 14
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label values, 14
list, 42
local, 11
log, 4
mean, 11
mvdecode, 4
normal, 11
oneway, 19, 23
predict, 21
prtest, 29, 30
prtesti, 31
pwcompare, 25
pwmean, 25
qnorm, 21
quietly, 23
ranksum, 12
recode, 14
robvar, 22
sqrt, 11
summarize, 6
tabodds, 43
tabstat, 18
tabulate, 27, 33–37
ttest, 8, 9, 23
twoway, 16
use, 4
webuse, 42
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