Standard error and confidence intervals (2.)

Report
UK FHS
Historical sociology
(2014)
Quantitative Data Analysis II.
Standard error and
confidence intervals (2.)
– for proportion
(and other parameters)
Jiří Šafr
jiri.safr(AT)seznam.cz
updated 25/11/2014
® Jiří Šafr, 2014
Content
• Principles of inferential statistics and interval
estimates (reminder, see http://metodykv.wz.cz/QDA2_CfI_1.ppt)
• Standard error (SE) and confidence interval
(CI) for categorical variables (p, %)
for numeric variables (mean) see http://metodykv.wz.cz/QDA2_CfI_1.ppt
• Assumptions for inferential statistics (SE, CI)
• How to compute CI for % in SPSS?
some alternatives
• Comparing for two population proportions
• Simultaneous confidence intervals
• Standard error and confidence intervals for other
parameters (proportion difference, correlation coefficient,
median,…)
2
For introduction into the logic of inferential statistics,
computation of standard error and confidence interval
for numerical variables, first see the presentation:
Standard error and confidence
intervals (1.)
– introduction to inferential statistics,
SE and CI for numerical variables (means)
http://metodykv.wz.cz/QDA2_CfI_1.ppt
Sampling Error
Population → sample → population
Random sampling error is encountered in survey research
because the sample selected is not a perfect representation
of the test population. [Assael, Keon 1982]
Vybírá se náhodně (bez
vracení) pouze jeden
výběrový soubor a údaje z
něho reprezentují základní
soubor (populaci).
Chybu způsobenou volbou
výběrového souboru lze s
určitou předem zvolenou
pravděpodobností vymezit
na základě
teorie výběrových šetření
4
Results from survey samples are always only
estimation of the true parameter (in population).
• Their accuracy is dependent mainly on sample size
and distribution of values (variance).
• Orientational tool: for large samples from large
(national) population, ca N=1000, the true
(population) relative frequencies (percent) range in
these intervals:
Source: [Special Eurobarometer 337]
However we will learn, how to compute it exactly and for whichever value and/or
parameter (level of measurement) (e.g. %, mean, % point difference, correlation, …)
5
Principle of inferential statistics – categorical variables
distribution of probability (i.e. %) in random sample(s) from population
[Figure 13.15 in De Vaus (1986)
2002: 304 or 232]
•
•
„If we have a random sample then probability theory again provides the answer. If we took a large number of random
samples most will come up with percentage estimates close to that which actually exists in the population. In only a
few samples will the sample estimates be way off the mark. In fact the sample estimates would approximate a
‘normal’ distribution (Figure 13.15).“ [De Vaus (1986) 2002: 232]
6 S
Na ose X je podíl (relativní počet výskytu) odpovědí pro volbu konzervativní strany v mnoha náhodných výběrech.
rostoucím počtem opakovaných náhodných výběrů se odhadovaná hodnota % blíží skutečné hodnotě v populaci.
Binomial distribution
Návštěva kostela
NSR, červenec–srpen 1956
Pravidelná
Nepravidelná
Málokdy
Nikdy
Celkem
%
30,3
24,6
28,6
16,5
100,0
Náhodný výběr 4000 osob, se rozdělí na
skupiny po 40 osobách, vznikne tak 100
dílčích náhodných výběrů. Toto
rozdělení odpovídá jako při dotazování u
100 reprezentativních průřezů. Tyto dílčí
náhodné výběry však nemají stejné
procento osob, které chodí do kostela jen
„málokdy“. Podle zákona velkých čísel
musí přitom menší odchylky vystupovat
častěji než velké. [Noelleová 1968: 115]
Podíl 27,5 % osob, které „málokdy“
navštěvuji kostel, tj. 11 ze 40
dotazovaných, vystupuje např. u 18 ze
100 dílčích náhodných výběrů, naproti
tomu jen v jednom výběru je podíl 10 %
= 4 ze 40 dotazovaných.
Z křivky zvonovitého tvaru lze vyčíst, jaké
rozdělení by se dalo očekávat v mezním
případě, kdyby se neprošetřovalo pouze
100, ale libovolné množství dílčích
náhodných výběrů.
7
What precedes computation of
confidence interval:
1. Standard error
And its calculation precedes computation of
0. variance/standard deviation
(2. level of confidence → z-values)
(general principle and how to obtain) → see
http://metodykv.wz.cz/QDA2_CfI_1.ppt
Standard error and estimation of
the parameter (e.g. mean or %)
• or generally standard error of a sample
• It quantifies uncertainty of our measuring
for mean: StD Error (of mean) SE =
for percent (%): StD Error (of proportion) SE =
• Note: Probability, i.e. proportion (%) is in fact a mean of
number of observations, so we calculate SE for proportion
essentially in the same way as SE of mean (standard
deviation of proportion divided by square root of sample
size).
9
Standard error
• Is smaller when sample size increases
(accuracy of parameter estimate increases)
• Increasing sample size twice, the confidence
interval decreases only 1,41 times
(√k-multiplicatively), that's why for twofold
accuracy we need quadruple sample size
• Usually there is sufficient accuracy of our results if
probability that ca 2/3 of measured values are within a
margin of mean or +/- 1 of their standard error (SE).
10
What standard error (SE) is for?
• It specifies, how (in)accurate are our results
• for omputation of confidence interval
• for testing, whether two (ore more)
parameters are different (in population)
• for testing, whether a sample parameter is
significantly (statistically) different from
zero in population (dělíme-li např. korelační
koeficient r jeho SE a dostaneme-li číslo větší než 2, pak je
s 95% pravděpodobností korelace nenulová, tj. existuje i v
celé populaci)
11
Confidence interval (assumptions)
• Further we will consider only Two-sided
confidence interval
(there is also one-sided CI, when we
determine only either Upper or Lower bound)
• for simple random sample
• and for large samples (n > 30)
• We assume at least normal distribution of
values of the phenomenon (which is in social
reality mostly on principle unrealistic)
12
Confidence Intervals
for qualitative data - nominal
variable
→ frequency (probability / percent)
% is probability multiplied by 100,
i.e. p 0,1 = 10 %
(thus p = 0,8 → 1 - p = 0,2)
Confidence interval for proportion (i.e. % / 100),
dichotomised variable
Point estimate ± choosen level of confidence
(statistical error) x Standard error of estimate
• Probability (point estimation) p = x/n
• Standard error of probability/proportion
SE = √ p(1 − p)/n
• Confidence Interval p ± zα/2(SE)
• C for 95 % error α = 0,05; zα/2 = 1,96
→ Existuje 95 % spolehlivost, že naměřená hodnota
ve výběru bude (v populaci) mezi hodnotami horní a
dolní hranice.
Máme-li proměnnou s více kategoriemi, pak počítáme p vždy jako
dichotomii té které kategorie oproti součtu ostatních
(např. vzdělání: VŠ / ostatní stupně (ZŠ+VY+SŠ).
14
Example: Voter turnout in 2006, CR
FREQ q34.
• To compute CI (SE) we he have to fill in
the formula and calculate it by hand (or
use some other tool)
Zdroj: data ISSP 2007
15
Example: Voter turnout in 2006, CR
• We have sample estimation (from ISSP 2007 survey) for
variable Voted in 2006 (catg. Voted / Didn‘t vote)
• Standard error of proportion (SE) for Voted:
– Probability of Voted = 750/1196 = 0,628 (= 62.8%)
– (Probability of Didn‘t vote = 446/1196 = 0.373)
• SE = √ 0.628(1 − 0.628)/1196 = 0.014
• True value of Voted estimated on our sample will be in
the interval (when p < 5 % → C=1.96):
0.628 ± 1.96 √ (0,628)(0,373)/1196 =
0.628 ± 0.0274 or (0,6006; 0.6554)
or 62.8 (± 2.7)%
Source: Data ISSP 2007, CR
16
Example: Voter turnout in 2006, CR
• Here we know exceptionally the true population
parameter from the official statistics: In the Elections to
the Chamber of Deputies held from 2 to 3 of June 2006,
64.47% Czech citizens participated. (official data from CZSO/ČSÚ)
• Our sample estimation (data ISSP 2007)
– for 95 % CI:
60.06 ← 62,8 → 65.54
– for 99 % CI (where zα/2 = 2.326)
59.60 ← 62,8 → 66,05
– for 90 % CI (where zα/2 = 1.645)
60.05 ← 62,8 → 65.01
Indeed, all interval estimates contain the true population proportion.
17
How to compute it in SPSS?
Not so easily as in case of
numerical variables (mean)
in SPSS CI for proportion (%) standardly
only in graph → BARCHART
GRAPH
/BAR(SIMPLE)=PCT BY
q34 /INTERVAL CI(95.0).
19
Zdroj: data ISSP 2007
BARCHART for % with CI, clicking in SPSS
20
Bivariate BARCHART (with CI for %)
• GRAPH /BAR(SIMPLE)=PCT BY q34 BY q38 /INTERVAL CI(95.0).
• For comparison of % „Voted in 2006 parliamentary election“ within
subgroups (e.g. along Union membership)
21
Zdroj: data ISSP 2007
1. In the output (on FREQ table) you can use (post)script
Script can be downloaded from: http://www.acrea.cz/sc_intervaly_spolehlivosti_cetnosti.htm
This is most convenient way. However it needs to be stored in a computer and you need the
appropriate version of the script fitting to your SPSS version, sometimes even some programming
environment needs to be installed (Python), and also it is probably only in Czech.
It doesn‘t exist in PSPP.
22
Source: data ISSP 2007, CR
2. Syntax routine CI for proportion [Pryce 2002]
http://www.spsstools.net/Syntax/Distributions/ProportionTestsAndCI.txt
Here we have to fill in results, e.g. from FREQ (univariate) or possibly CROSSTAB (bivariate). In fact there are four tests in this syntax.
For univariate description it is the second test Large-Sample Confidence Interval for a Single Population
Proportion. Fill in only values of n a p, you can also choose CI (originaly set to 99% CI) and decimals shown.
*-------------------------------------------------------------------------------.
*-------------------------------------------------------------------------------.
* Large-Sample Confidence Interval for a Single Population Proportion.
* (see Moore and McCabe (2001) Intro to the Practice of Statistics, p. 586 -588).
*-------------------------------------------------------------------------------.
*For the inverse normal computation, I use the approximation used by http://www.hpmuseum.org/software/67pacs/67ndist.htm adap ted from Abramowitz and Stegun,
Handbook of Mathematical Functions, National Bureau of Standards 1970.
MATRIX.
COMPUTE n = {4040}. /* Enter the sample size here (change the number in curly brackets)*/
COMPUTE x = {2048}. /* Enter the number of "successes" (change the number in curly brackets)*/
COMPUTE CONFID = {0.99}. /* Enter the desired confidence level here */
*The remainder of the syntax calculates the Confidence Interval given the values for n and x which you have entered above.
*NB you don't need to alter anything from here on.
COMPUTE Q = 0.5 * (1-CONFID). COMPUTE A = ln(1/(Q**2)).
COMPUTE T_ = SQRT(A).
COMPUTE zstar = T_ - ((2.515517 + (0.802853*T_) + (0.010328*T_**2))/ (1 + (1.432788*T_) + (0.189269*T_**2) + (0.001308*T_**3))).
COMPUTE phat = x/n.
COMPUTE SE_phat = SQRT((phat*(1-phat))/n). COMPUTE m = zstar * SE_phat.
COMPUTE LOWER = phat - m. COMPUTE UPPER = phat + m.
COMPUTE ANSWER = {n, phat, zstar, SE_phat, Lower, Upper}. PRINT ANSWER / FORMAT "F10.5" /Title = "Confidence Interval for a Single Population Proportion" /
CLABELS = n, phat, zstar, SE, Lower, Upper.
END MATRIX.
*NB if you want to obtain values to a greater (lesser) number of decimal places, change the format specified in the last but one line of the syntax.
*e.g. if you want only 3 decimal places, change the format to "F10.3".
*------------------------------------------------------------------------------.
*------------------------------------------------------------------------------.
The output:
Run MATRIX procedure:
Confidence Interval for a Single Population Proportion
n
phat
zstar
SE
Lower
1196,000
,627
1,960
,014
,600
------ END MATRIX ----And don't forget, if you use this script (e.g. in diploma thesis) you should credit it, cite: Gwilym Pryce 2002. Large-Sample Confidence Interval for a Single
Population Proportion. Inference for Proportions. Available at: http://www.spsstools.net/Syntax/Distributions/ProportionTestsAndCI.txt.
Upper
,655
23
Source: data ISSP 2007, CR
For contingency table in SPSS (only) graphically or
computing via Syntax routine CI for proportion [Pryce 2002]
Example: Type of housing [s31] by description of place of living (size of community) [s21]
CROSS s31 BY s21.
And fill results in
into the formula
or the syntax
routine by Pryce [2002].
Source: data ISSP 2007, CR
• for category „small town“:
Rodinný domek
Menší bytový dům
Větší bytový dům
p dolní mez horní mez
0,3266
0,2805
0,3727
0,1482
0,1133
0,1832
0,5251
0,4761
0,5742
CROSS s31 BY s21 /CEL COL.
GRAPH /BAR(SIMPLE)=PCT BY
s31 BY s21/INTERVAL CI(95.0).
24
Web calculators of confidence
interval for nominal variables (%)
• http://ncalculators.com/statistics/confidence-interval-calculator.htm
http://www.surveysystem.com/sscalc.htm
• http://vassarstats.net/prop1.html
• Confidence Interval for the Difference Between Two Independent
Proportions.
http://vassarstats.net/prop2_ind.html
25
Orientational tool (if there is no computer nor calculator):
Statistical margins pro binomial distribution
Value of 2σ — two standard
deviations — in %
→ Level of statistical
significance 95,45 %
n = sample size (random sample)
p = frequency in population (%)
Source: [Noelle 1968: 118]
26
Task
• Compute confidence interval for proportion
of people with university diploma in CR
using sample data ISSP 2007.
• Compare it with the true population value
(statistics of CZSO/ČSÚ for 2007).
• What‘s wrong?
→ show solution in AKD2_1_CfI_RESENI
27
Comparing for two population proportions
(dichotomised variables in crosstabulation)
• We can compute confidence interval for proportion of
specific value/category within subgroups or for already
existing results. For example, dichotomised variables: Voted
(dependent var) along categories of Religion (Christian/otherwise)
(independent var) and to compare, whether interval estimates within
categories of Religion overlap or not.
• More exact and easier it is via computing
CF of % difference between the proportions/categories
• If the confidence interval of the proportion difference is
not including 0 (i.e. it is not „zero“ within the whole
population), we can assert, that % difference between the
(sub)categories is statistically significant (at given p), i.e. it
holds true with given statistical error for whole population.
→ You can compute it by hand (for formula see later) or using SPSS syntax routine by G.
Pryce [2002] http://www.spsstools.net/Syntax/Distributions/ProportionTestsAndCI.txt
use the last (4.) test Large-sample Confidence Intervals for Comparing
for two population proportions.
• This method can be applied to a crosstabulation with more categories 28
→ step by step focusing on one by one value/category comparison.
Comparing for two population proportions
SPSS syntax routine by G. Pryce [2002]
http://www.spsstools.net/Syntax/Distributions/ProportionTestsAndCI.txt
• Here we have to fill in results, e.g. from FREQ (univariate) or possibly
CROSSTAB (bivariate). In fact there are four tests in this syntax.
• For comparing for two population proportions it is the fourth test Largesample Confidence Intervals for Comparing for two population
proportions. Fill in only values of n1, n2 and p1, p2, you can also
choose CI (originally set to 90% CI) and decimals shown.
*-------------------------------------------------------------------------------.
*-------------------------------------------------------------------------------.
* Large-sample Confidence Intervals for Comparing for two population proportions.
* (see Moore and McCabe (2001) Intro to the Practice of Statistics, p. 602-604).
*-------------------------------------------------------------------------------.
*For the inverse normal computation, I use the approximation used by
http://www.hpmuseum.org/software/67pacs/67ndist.htm adapted from Abramowitz and Stegun,
Handbook of Mathematical Functions, National Bureau of Standards 1970.
Example: Non-participation in
MATRIX.
COMPUTE n1 = {1222}. /* Enter the first sample size here (change the number in curly brackets)*/
COMPUTE n2 = {1222}. /* Enter the second sample size here (change the number in curly brackets)*/
COMPUTE x1 = {958}. /* Enter the number of "successes" for sample 1 here (change the nb in curly brackets)*/
COMPUTE x2 = {1016}. /* Enter the number of "successes" for sample 2 here (change the nb in curly brackets)*/
COMPUTE CONFID = {0.95}. /* Enter the desired confidence level here */
*The remainder of the syntax calculates the Confidence Interval given the values for n and
x which you have entered above.
*NB you don't need to alter anything from here on.
COMPUTE Q = 0.5 * (1-CONFID).
COMPUTE A = ln(1/(Q**2)).
COMPUTE T_ = SQRT(A).
COMPUTE zstar = T_ - ((2.515517 + (0.802853*T_) + (0.010328*T_**2))/
(1 + (1.432788*T_) + (0.189269*T_**2) + (0.001308*T_**3))).
COMPUTE p1hat = x1/n1.
COMPUTE p2hat = x2/n2.
COMPUTE SE_phat = SQRT(((p1hat*(1-p1hat))/n1) + (p2hat*(1-p2hat))/n2)).
COMPUTE m = zstar * SE_phat.
COMPUTE LOWER = (p1hat - p2hat) - m.
COMPUTE UPPER = (p1hat - p2hat) + m.
COMPUTE diffp1p2 = p1hat - p2hat.
COMPUTE ANSWER = {n1, n2, diffp1p2, zstar, SE_phat, Lower, Upper}.
PRINT ANSWER / FORMAT "F10.5"
/Title = "Confidence Interval for Comparing 2 Proportions"
/ CLABELS = n1, n2, diffp1p2, zstar, SE, Lower, Upper.
END MATRIX.
Sport (q13_a) = 958
Culture: (q13_b) = 1016
TOTAL = 1222.
The output:
Sport clubs and Culture association
[ISSP 2007, CR]
The result: the CI is not crossing
0 → the difference 4,7 % points is
statistically significant (at p <
5%).
Run MATRIX procedure:
Confidence Interval for Comparing 2 Proportions
n1
n2
diffp1p2
zstar
1222,00000 1222,00000
-,04746
1,96039
------ END MATRIX -----
SE
,01592
Lower
-,07866
Upper
-,01626
29
And don't forget, if you use this script (e.g. in diploma thesis) you should credit it, cite: Gwilym Pryce 2002. Large-Sample Confidence Interval for a Single
Population Proportion. Inference for Proportions. Available at: http://www.spsstools.net/Syntax/Distributions/ProportionTestsAndCI.txt.
Or you can use
Web Calculator for Confidence Interval for
the Difference Between Two
Independent Proportions
http://vassarstats.net/prop2_ind.html
30
Simultaneous confidence intervals
(for proportion) → multiple comparison problem
• So far we have made independent conclusions, but
if we want to assess several proportions together,
we need to assure that all parameters were covered by
predetermined desired level of confidence.
• For Simultaneous conclusion about several proportions
we make the overall confidence C stricter to z α / S
where S = number of proportions for which we need
simultaneous confidence intervals
• For example: for 4 proportions, with desired α = 0,05
z α / 4 = z α / 0,0125 = 0,02497 i.e. rounded to C = 2,5
(we use it instead of common C = 1,96 for a single population proportion)
This is similar method to multiple comparison of means in more than two
subgroups, i.e. Post-hoc test in Analysis of variance.
See statistical tables with critical values of standard normal test for simultaneous
testing.
Source: [Řehák, Řeháková 1986: 64-65]
31
Confidence interval v SPSS ?
• SPSS computes CI only for numerical variables,
i.e. mean (e.g. EXPLORE)
• (In OLS regression we can get CI for regression
coefficient B, in logistic regression for exp(B).)
• However, it is easy to compute standard error (e.g.
for proportion % or correlation coefficient;
sometimes SPSS provides it) and filling it into the
formulas, we can calculate CI easily by hand (see later)
• Alternatively we can use syntax routines, e.g. for
proportion by G. Pryce [2002]
http://www.spsstools.net/Syntax/Distributions/ProportionTestsAndCI.txt
or scripts with post-hoc adaptation in outpout (for
•
univariate % http://www.acrea.cz/skripty-interval-spolehlivosti-cetnosti.htm)
Or we can calculate CI outside SPSS, e.g. using the internet calculators… (see
later)
32
Standard error and Confidence
intervals for various parameters
(correlation coefficient, median,
difference of proportion (%), …)
Standard error and CI of
correlation coefficient (in SPSS)
SE is not included within CORRELATION but it is in CROSSTABS
CROSSTABS OC2011 BY PrijmD
/FORMAT=NOTABLES /STATISTICS=CORR .
CI (95%) for R = 0,072 ± 1,96*0,023 = 0,072 ± 0,045 or 0,027 ← 0,072 → 0,117
CI correlation coefficient can be computed at http://vassarstats.net/rho.html
34
Computation of standard error
• for mean
• for standard deviation
• for median
• pro correlation coefficient
or
35
Computation of standard error
• for proportion (%)
SE = √ p(1 − p) / n
• for difference of proportion (%) p1- p2
Web Calculator for Confidence Interval for the Difference
Between Two Independent Proportions
http://vassarstats.net/prop2_ind.html
• for Odds Ratio
More on http://davidmlane.com/hyperstat/A111955.html
http://www.miislita.com/information-retrieval-tutorial/a-tutorial-on-standard-errors.pdf
36
Routines for Confidence intervals
in SPSS syntax
• for proportion (%)
http://www.spsstools.net/Syntax/Distributio
ns/ProportionTestsAndCI.txt
• for median
http://www.spsstools.net/Syntax/Distributio
ns/Calculate95PercCIforTheMedian.txt
37

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