Ambitious title?
Confidence intervals, design effects
and significance tests for surveys.
How to calculate sample numbers
when planning a survey.
Scot Exec Course Nov/Dec 04
Summary
• Statistical inference
– Design based
– Model based
• Confidence intervals and hypothesis tests general
• Their modification for survey designs
– Design effects and design factors
• Calculation of sample numbers for studies
– Their modification for complex surveys
Scot Exec Course Nov/Dec 04
Statistical inference
• Making inferences about some aspect of the
population, using observation to draw
conclusions about the population now, or
will evolve in future
• Data are what we are given
• Inference allows us to turn them into
information
Scot Exec Course Nov/Dec 04
Elements needed for statistical
inference – design based
• Want to learn something about a population
• You have
– A model of how the sample was selected from the
population.
– Some data obtained from the sample
– Knowledge of how to estimate!
• E.g. Obtain data on the income of 10,000 from a population of 5
million.
• Need inference to estimate the income distribution of the whole 5
million and to know how close this is to the population value
Scot Exec Course Nov/Dec 04
Elements needed for statistical
inference – model based
• You have
– A model that could have generated the data for your
population, along with ideas about what current and
future populations this might generalise to..
– Some data that can be assumed to be generated by this
model.
– Knowledge of how to carry out the inference!
• E.g. Obtain data on the income of 10,000 from a population and can
make the assumption that the income distribution follows some
mathematical distribution
• Need inference about the assumed model for the income distribution of
the whole 5 million and how close your estimate will be to the true
value
Scot Exec Course Nov/Dec 04
How do design and model
based inferences differ?
•
•
•
•
Conceptually poles apart
In practice they give the same answers
Except when numbers are small
Or when a large proportion of the
population has been sampled
• But its good to think about what you are
doing and decide which type fits your
problem
Scot Exec Course Nov/Dec 04
Next set of results
• Apply to a simple unstructured sample
– No clustering
– No stratification
– No weighting
• Taken from a population with replacement (not a
problem in model based inference)
• Exactly the same large-sample results apply for
model-based and design-based inferences
Scot Exec Course Nov/Dec 04
Mean of 9 x s
x
x
?m
m?
Scot Exec Course Nov/Dec 04
st .dev .

st.dev. 
n
Standard error of the mean
x  m
Approx a normal distr with s.d.

n
The data are fixed, so this tells us where m is likely to be.

n
is called the standard error of the sample mean
Sometimes s.e.mean - it measures the expected distance of
the “true” mean from the mean of the observed sample.
A 100(1a)% confidence interval for m from the
normal distribution Is
x
 za/2 s.e.m.
Scot Exec Course Nov/Dec 04
Values of Z for confidence
intervals
• 95% c.I. Gives Z = 1.96
• 99%
Z = 2.58
• 68%
Z=1
• 90%
Z = 1.64
Scot Exec Course Nov/Dec 04
We can use it for proportions too
• Want too estimate a proportion p - e.g. a
proportion of 20 year olds who use the internet
–Then r/n estimates p
p (1  p ) / n
–to use this formula we replace p with ( pˆ  r n )
–with standard error
•A rule of thumb is that this approximation is OK
if the smaller of r and (n-r) is >5.
Scot Exec Course Nov/Dec 04
Are these formulae good enough?
• Yes – unless your survey is too small to be
any use
• They extend easily to differences in means
and proportions
• Similar approximate results apply to
regression models and logistic regressions
• BUT – they only apply to simple samples
Scot Exec Course Nov/Dec 04
But my data are more complicated than this
And nobody will let me put standard erorrs or
confidence intervals in my report
• A goal of a good statistical report is that it should
not include and tables or graphs where what seems
to be information are just the result of chance
variation (noise).
– set out your task in terms of an outcome predicted from
other factors
– Carry out a set of regression predictions
– Base the tables to go in the report on the regression
models that are found to be more than chance effects
Scot Exec Course Nov/Dec 04
Inferences for complex surveys
• The usual formulae and regression models
don’t hold
• Most surveys use weighting
• And allowances for clustering and
stratification have to be made
• Software that modifies the results we have
just discussed and calculates them correctly
for complex surveys is now available
Scot Exec Course Nov/Dec 04
Two main methods are used
• Taylor linearisation – theory of this all
worked out in the 1940s and 50s
• Replication methods, jacknives and
bootsraps – 1960s and 1970s
• Only now is software readily available to do
things properly
Scot Exec Course Nov/Dec 04
Getting by without the correct software
• Carry out an analysis using an ordinary computer
package (eg. SAS, SPSS simple procedures)
• But use a weight in the analysis to get results that
will correct the bias in the estimates
• Your weighted analysis will get you the wrong
standard errors and wrong tests, but the estimates
will be about right.
• Use design effect tables to get some idea of the
standard errors
Scot Exec Course Nov/Dec 04
Using the correct software
• Is not difficult – PEAS web site explains how
• Routines are available in SAS, SPSS, STATA and
R
• But it does mean that you need to get details of the
survey design
• E.g. PSU, stratification variables need to be
available
• Easier for you than for me
Scot Exec Course Nov/Dec 04
Getting by without the correct software
• Use a table of design effects (DE)
• Often published with the surveys
• To get a s.e. from a complex survey
– Calculate the design factor (DF) as the square root of
the DE
• Multiply the s.e. from a simple analysis by DF
• For most household surveys DEs vary from about
0.8 to 2 or 3.
• This is a rough and ready method and will only
work if weights are not too far from 1.0
Scot Exec Course Nov/Dec 04
Disadvantages of this
• DEs are not constant for a survey
• They are also different (usually lower) when
subgroups of a survey are selected
• They may also be lower in complicated
models, like regressions where it is also
very hard to know how to apply them.
• Methods are approximate
Scot Exec Course Nov/Dec 04
Uses of design effects (DEs)
• They tell you about how well your survey
design has worked
• Most survey software produce estimates of
design effects with their output
• A design effect of 2 means your effective
sample size is halved
• It is good to have such estimates when
planning sample numbers for surveys.
Scot Exec Course Nov/Dec 04
Sample numbers for planning
studies
• Think ahead about the sort of comparisons
you might want to make
• Are you interested in time trends?
• Or in comparisons between certain groups
– If so, what proportions in each
• Do you want to estimate something (eg %
of children in poverty)?
Scot Exec Course Nov/Dec 04
Use spread sheet sample
numbers.xls
Scot Exec Course Nov/Dec 04
To modify these for surveys
• Simply multiply your answer by an estimate
of the design effect
• Or try to do the next survey better by
getting a smaller design effect
Scot Exec Course Nov/Dec 04