Sumber : http://www.epa.gov/nheerl/arm/surdesignfaqs.htm
A sample is any subset of the target population, i.e., any collection of its elements. Sampling
methods may be classified into either probability-based sample methods or non-probabilitybased sampling methods. Probability-based methods are discussed in subsequent sections.
Non-probability methods include chunk samples, expert choice samples, and quota samples.
Chunk Samples. Scientists often draw conclusions using an arbitrary or fortuitous collection
of sites. The sites are gathered haphazardly or "happen to be handy." Often the scientist
implicitly assumes that the sites are typical for a larger universe of sites about which
conclusions are desired. Such an assumption has only the individual's judgment as a basis
and can not be easily defended. The sites are an unknown "chunks" of the target population
and consequently no basis exists to make a scientific inference to the target population
without invoking assumptions that can not be verified.
Expert Choice Samples. Expert choice sampling is a form of judgment sampling that is a
more developed form of non-random selection. An expert, or experts, may define a set of
criteria to be met for a site to be included in the sample. Not all sites that meet the criteria are
included. Criteria usually result in the designation of "typical" sites for the study. A fairly good
sample may result given that the expert was skillful in defining the criteria and locating typical
sites that met the criteria. However there is no way to be sure. A different expert would
probably use different criteria or pick different sites that et the criteria. Without invoking
additional assumptions, no basis exists to make inferences to the target population and know
the uncertainty associated with the inference.
Quota Samples. Quota sampling is commonly used in market research. The target
population is divided according to one or more characteristics, e.g., age, sex, and geographic
area. For two age groups, two sexes, and three geographic areas, a total of 12 population
cells are defined. The cells are similar to strata in stratified random sampling. A quota sample
then contains a pre-determined number of individuals in each of the 12 cells. The interviewer
then simply "fills the quota" for each cell. The individuals may be the first individuals
encountered or the interviewer may have the option of using judgment in selecting the
individuals. The sample of individuals in a cell is either a chunk sample or a judgment
sample. Individuals may refuse or be unavailable, but new individuals are contacted until the
quota of individuals is achieved. However, the problem of selection bias due to non-response
still remains. Hence as before there is no basis for an inference to the target population.
How many sample sites to use?
The most commonly asked question is: How many sample sites do I need? This is an
important question as it directly determines the precision of any statement derived from the
sample data. An answer requires detailed information on all the estimates that will be
produced from the survey, the precision desired for each estimate, and knowledge of the
variability expected. A reality faced in most studies is that the number of objectives creates a
need for many more sample sites than budget and operational constraints allow.
Consequently, the total number of sites in many situations is known from these constraints
and the question is which objectives are the most important. It is usual to have some subobjectives dropped due to sample size limitations.
Sample size calculations are available in most survey sampling textbook and will not be
discussed here. One specific situation of interest is when the objectives call for the estimation
of a proportion, e.g., proportion of stream length that meets a designated use. In this case,
sample size calculations depend only on the proportion, precision required, and confidence
required. Approximate precision estimates for proportions can be obtained by assuming the
survey designs are simple random samples. Under this condition the estimated precision can
be estimated using procedures given by Cochran (1987) for proportions. Go to Top
Precision, as a percent, is determined from precision = Z1- * 100 * Sqrt[ p(1-p)/n]
To calculate precision requires knowledge of p, the proportion to be estimated. However, a
conservative estimate of precision can be obtained by assuming p to be 0.5, which gives the
maximum variance. Z1- is related to the level of confidence required for the estimate. If
desire 90% confidence, then use 1.645. If desire 95% confidence, then use 2.
Table 1. Precision to achieve 90% confidence in estimates of selected proportions.
Assumed
Precision with 90% Confidence
Precision with 95% Confidence
for alternative sample sizes
for alternative sample sizes
Proportion
(percent)
25
50
100
400
1000
25
50
100
400
1000
20%
±13
±9
±7
±3
±2
±16
±11
±8
±4
±3
50%
±17
±12
±8
±4
±3
±20
±13
±10
±5
±3
If the survey designs are actually based on the spatially-restricted survey designs, the
actual precision estimates are expected to be lower (better) than those stated. Go to Top
How do Sites get selected?
Several processes lead to the selection of sites. The first process identifies the
resource characteristics and target population and results in a sample frame that
contains all sites within the target population. The second process establishes a
spatial grid and hierarchical structure that result in cells containing single, or a small
number/area of sites. These two results are then combined resulting in each site, or
small number/area of sites assigned a hierarchical cell address. Randomization and
statistical weighting processes produce a sequence of all sites from which a
systematic random sample is selected.
Sumber : http://home.ubalt.edu/ntsbarsh/stat-data/Surveys.htm
Pertemuan :19,20,21,22,23,24
Introduction
The main idea of statistical inference is to take a random sample from a population
and then to use the information from the sample to make inferences about particular
population characteristics such as the mean (measure of central tendency), the
standard deviation (measure of spread) or the proportion of units in the population
that have a certain characteristic. Sampling saves money, time, and effort.
Additionally, a sample can, in some cases, provide as much information as a
corresponding study that would attempt to investigate an entire population-careful
collection of data from a sample will often provide better information than a less
careful study that tries to look at everything.
We must study the behavior of the mean of sample values from different specified
populations. Because a sample examines only part of a population, the sample mean
will not exactly equal the corresponding mean of the population. Thus, an important
consideration for those planning and interpreting sampling results, is the degree to
which sample estimates, such as the sample mean, will agree with the corresponding
population characteristic.
In practice, only one sample is usually taken (in some cases such as "survey data
analysis" a small "pilot sample" is used to test the data-gathering mechanisms and to
get preliminary information for planning the main sampling scheme). However, for
purposes of understanding the degree to which sample means will agree with the
corresponding population mean, it is useful to consider what would happen if 10, or
50, or 100 separate sampling studies, of the same type, were conducted. How
consistent would the results be across these different studies? If we could see that the
results from each of the samples would be nearly the same (and nearly correct!), then
we would have confidence in the single sample that will actually be used. On the
other hand, seeing that answers from the repeated samples were too variable for the
needed accuracy would suggest that a different sampling plan (perhaps with a larger
sample size) should be used.
Quota Sampling: Quota sampling is availability sampling, but with the constraint
that proportionality by strata be preserved. Thus the interviewer will be told to
interview so many white male smokers, so many black female nonsmokers, and so
on, to improve the representatives of the sample. Maximum variation sampling is a
variant of quota sampling, in which the researcher purposively and non-randomly
tries to select a set of cases, which exhibit maximal differences on variables of
interest. Further variations include extreme or deviant case sampling or typical case
sampling.
What Is the Margin of Error
Estimation is the process by which sample data are used to indicate the
value of an unknown quantity in a population.
Results of estimation can be expressed as a single value; known as a
point estimate, or a range of values, referred to as a confidence
interval.
Whenever we use point estimation, we calculate the margin of error
associated with that point estimation. For example; for the estimation
of the population proportion, by the means of sample proportion (P),
the margin of errors calculated often as follows:
±1.96 [P(1-P)/n]1/2
In newspapers and television reports on public opinion pools, the
margin of error is often appears in small font at the bottom of a table
or screen, respectively. However, reporting the amount of error only,
is not informative enough by itself, what is missing is the degree of
the confidence in the findings. The more important missing piece of
information is the sample size n. that is, how many people participated
in the survey, 100 or 100000? By now, you know it well that the
larger the sample size the more accurate is the finding, right?
The reported margin of error is the margin of "sampling error". There
are many nonsampling errors that can and do affect the accuracy of
polls. Here we talk about sampling error. The fact that subgroups have
larger sampling error than one must include the following statement:
"Other sources of error include but are not limited to, individuals
refusing to participate in the interview and inability to connect with
the selected number. Every feasible effort is made to obtain a response
and reduce the error, but the reader (or the viewer) should be aware
that some error is inherent in all research."
If you have a yes/no question in a survey, you probably want to
calculate a proportion P of Yes's (or No's). Under simple random
sampling survey, the variance of P is P(1-P)/n, ignoring the finite
population correction, for large n, say over 30. Now a 95% confidence
interval is
P - 1.96 [P(1-P)/n]1/2, P + 1.96 [P(1-P)/n]1/2.
A conservative interval can be calculated, since P(1-P) takes its
maximum value when P = 1/2. Replace 1.96 by 2, put P = 1/2 and you
have a 95% consevative confidence interval of 1/n1/2. This
approximation works well as long as P is not too close to 0 or 1. This
useful approximation allows you to calculate approximate 95%
confidence intervals.
References and Further Readings:
Casella G., and R. Berger, Statistical Inference, Wadsworth Pub. Co., 2001.
Kish L., Survey Sampling, Wiley, 1995.
Lehmann E., and G. Casella, Theory of Point Estimation, Springer Verlag, New York, 1998.
Levy P., and S. Lemeshow, Sampling of Populations: Methods and Applications, Wiley, 1999.
Sample Size Determination
The question of how large a sample to take arises early in the planning
of any survey. This is an important question that should be treated
lightly. To take a large sample than is needed to achieve the desired
results is wasteful of resources whereas very small samples often lead
to that are no practical use of making good decision. The main
objective is to obtain both a desirable accuracy and a desirable
confidence level with minimum cost.
Pilot Sample: A pilot or preliminary sample must be drawn from the
population and the statistics computed from this sample are used in
determination of the sample size. Observations used in the pilot
sample may be counted as part of the final sample, so that the
computed sample size minus the pilot sample size is the number of
observations needed to satisfy the total sample size requirement.
People sometimes ask me, what fraction of the population do you
need? I answer, "It's irrelevant; accuracy is determined by sample size
alone" This answer has to be modified if the sample is a sizable
fraction of the population.
For an item scored 0/1 for no/yes, the standard deviation of the item
scores is given by SD = [p(1-p)/N] 1/2 where p is the proportion
obtaining a score of 1, and N is the sample size.
The standard error of estimate SE (the standard deviation of the range
of possible p values based on the pilot sample estimate) is given by
SE= SD/N½. Thus, SE is at a maximum when p = 0.5. Thus the worst
case scenario occurs when 50% agree, 50% disagree.
The sample size, N, can then be expressed as largest integer less than
or equal to 0.25/SE2
Thus, for SE to be 0.01 (i.e. 1%), a sample size of 2500 would be
needed; 2%, 625; 3%, 278; 4%, 156, 5%, 100.
Note, incidentally, that as long as the sample is a small fraction of the
total population, the actual size of the population is entirely irrelevant
for the purposes of this calculation.
Sample sizes with regard to binary data:
n = [t2 N p(1-p)] / [t2 p(1-p) + 2 (N-1)]
with N being the size of the total number of cases, n being the sample
size,  the expected error, t being the value taken from the t
distribution corresponding to a certain confidence interval, and p being
the probability of an event.
For a finite population of size N, the standard error of the sample mean
of size n, is:
[(N -n)/(nN)]½
There are several formulas for the sample size needed for a t-test. The
simplest one is
n = 2(Z+Z)22/D2
which underestimates the sample size, but is reasonable for large
sample sizes. A less inaccurate formula replaces the Z values with t
values, and requires iteration, since the df for the t distribution
depends on the sample size. The accurate formula uses a non-central t
distribution and it also requires iteration.
The simplest approximation is to replace the first Z value in the above
formula with the value from the studentized range statistic that is used
to derive Tukey's follow-up test. If you don't have sufficiently detailed
tables of the studentized range, you can approximate the Tukey
follow-up test using a Bonferroni correction. That is, change the first Z
value to Z where k is the number of comparisons.
Neither of these solutions is exact and the exact solution is a bit messy.
But either of the above approaches is probably close enough,
especially if the resulting sample size is larger than (say) 30.
A better stopping rule for conventional statistical tests is as follows:
Test some minimum (pre-determined) number of subjects.
Stop if p-value is equal to or less than .01, or p-value equal to or
greater than .36; otherwise, run more subjects.
Obviously, another option is to stop if/when the number of subjects
becomes too great for the effect to be of practical interest. This
procedure maintains  about 0.05.
We may categorized probability proportion to size (PPS) sampling,
stratification, and ratio estimation (or any other form of model assisted
estimation) as tools that protect one from the results of a very unlucky
sample. The first two (PPS sampling and stratification) do this by
manipulation of the sampling plan (with PPS sampling conceptually a
limiting case of stratification). Model assisted estimation methods
such as ratio estimation serve the same purpose by introduction of
ancillary information into the estimation procedure. Which tools are
preferable depends, as others have said, on costs, availability of
information that allows use of these tools, and the potential payoffs
(none of these will help much if the stratification/PPS/ratio estimation
variable is not well correlated with the response variable of interest).
There are also heuristic methods for determination of sample size. For
example, in healthcare behavior and process measurement sampling
criteria are designed for a 95% CI of 10 percentage points around a
population mean of 0.50; There is a heuristic rule: "If the number of
individuals in the target population is smaller than 50 per month,
systems do not use sampling procedures but, attempt to collect data
from all individuals in the target population."
Sumber : http://home.ubalt.edu/ntsbarsh/stat-data/Surveys.htm
Introduction
The main idea of statistical inference is to take a random sample from a population
and then to use the information from the sample to make inferences about particular
population characteristics such as the mean (measure of central tendency), the
standard deviation (measure of spread) or the proportion of units in the population
that have a certain characteristic. Sampling saves money, time, and effort.
Additionally, a sample can, in some cases, provide as much information as a
corresponding study that would attempt to investigate an entire population-careful
collection of data from a sample will often provide better information than a less
careful study that tries to look at everything.
We must study the behavior of the mean of sample values from different specified
populations. Because a sample examines only part of a population, the sample mean
will not exactly equal the corresponding mean of the population. Thus, an important
consideration for those planning and interpreting sampling results, is the degree to
which sample estimates, such as the sample mean, will agree with the corresponding
population characteristic.
In practice, only one sample is usually taken (in some cases such as "survey data
analysis" a small "pilot sample" is used to test the data-gathering mechanisms and to
get preliminary information for planning the main sampling scheme). However, for
purposes of understanding the degree to which sample means will agree with the
corresponding population mean, it is useful to consider what would happen if 10, or
50, or 100 separate sampling studies, of the same type, were conducted. How
consistent would the results be across these different studies? If we could see that the
results from each of the samples would be nearly the same (and nearly correct!), then
we would have confidence in the single sample that will actually be used. On the
other hand, seeing that answers from the repeated samples were too variable for the
needed accuracy would suggest that a different sampling plan (perhaps with a larger
sample size) should be used.
Quota Sampling: Quota sampling is availability sampling, but with the constraint
that proportionality by strata be preserved. Thus the interviewer will be told to
interview so many white male smokers, so many black female nonsmokers, and so
on, to improve the representatives of the sample. Maximum variation sampling is a
variant of quota sampling, in which the researcher purposively and non-randomly
tries to select a set of cases, which exhibit maximal differences on variables of
interest. Further variations include extreme or deviant case sampling or typical case
sampling.
What Is the Margin of Error
Estimation is the process by which sample data are used to indicate the
value of an unknown quantity in a population.
Results of estimation can be expressed as a single value; known as a
point estimate, or a range of values, referred to as a confidence
interval.
Whenever we use point estimation, we calculate the margin of error
associated with that point estimation. For example; for the estimation
of the population proportion, by the means of sample proportion (P),
the margin of errors calculated often as follows:
±1.96 [P(1-P)/n]1/2
In newspapers and television reports on public opinion pools, the
margin of error is often appears in small font at the bottom of a table
or screen, respectively. However, reporting the amount of error only,
is not informative enough by itself, what is missing is the degree of
the confidence in the findings. The more important missing piece of
information is the sample size n. that is, how many people participated
in the survey, 100 or 100000? By now, you know it well that the
larger the sample size the more accurate is the finding, right?
The reported margin of error is the margin of "sampling error". There
are many nonsampling errors that can and do affect the accuracy of
polls. Here we talk about sampling error. The fact that subgroups have
larger sampling error than one must include the following statement:
"Other sources of error include but are not limited to, individuals
refusing to participate in the interview and inability to connect with
the selected number. Every feasible effort is made to obtain a response
and reduce the error, but the reader (or the viewer) should be aware
that some error is inherent in all research."
If you have a yes/no question in a survey, you probably want to
calculate a proportion P of Yes's (or No's). Under simple random
sampling survey, the variance of P is P(1-P)/n, ignoring the finite
population correction, for large n, say over 30. Now a 95% confidence
interval is
P - 1.96 [P(1-P)/n]1/2, P + 1.96 [P(1-P)/n]1/2.
A conservative interval can be calculated, since P(1-P) takes its
maximum value when P = 1/2. Replace 1.96 by 2, put P = 1/2 and you
have a 95% consevative confidence interval of 1/n1/2. This
approximation works well as long as P is not too close to 0 or 1. This
useful approximation allows you to calculate approximate 95%
confidence intervals.
References and Further Readings:
Casella G., and R. Berger, Statistical Inference, Wadsworth Pub. Co., 2001.
Kish L., Survey Sampling, Wiley, 1995.
Lehmann E., and G. Casella, Theory of Point Estimation, Springer Verlag, New York, 1998.
Levy P., and S. Lemeshow, Sampling of Populations: Methods and Applications, Wiley, 1999.
Sample Size Determination
The question of how large a sample to take arises early in the planning
of any survey. This is an important question that should be treated
lightly. To take a large sample than is needed to achieve the desired
results is wasteful of resources whereas very small samples often lead
to that are no practical use of making good decision. The main
objective is to obtain both a desirable accuracy and a desirable
confidence level with minimum cost.
Pilot Sample: A pilot or preliminary sample must be drawn from the
population and the statistics computed from this sample are used in
determination of the sample size. Observations used in the pilot
sample may be counted as part of the final sample, so that the
computed sample size minus the pilot sample size is the number of
observations needed to satisfy the total sample size requirement.
People sometimes ask me, what fraction of the population do you
need? I answer, "It's irrelevant; accuracy is determined by sample size
alone" This answer has to be modified if the sample is a sizable
fraction of the population.
For an item scored 0/1 for no/yes, the standard deviation of the item
scores is given by SD = [p(1-p)/N] 1/2 where p is the proportion
obtaining a score of 1, and N is the sample size.
The standard error of estimate SE (the standard deviation of the range
of possible p values based on the pilot sample estimate) is given by
SE= SD/N½. Thus, SE is at a maximum when p = 0.5. Thus the worst
case scenario occurs when 50% agree, 50% disagree.
The sample size, N, can then be expressed as largest integer less than
or equal to 0.25/SE2
Thus, for SE to be 0.01 (i.e. 1%), a sample size of 2500 would be
needed; 2%, 625; 3%, 278; 4%, 156, 5%, 100.
Note, incidentally, that as long as the sample is a small fraction of the
total population, the actual size of the population is entirely irrelevant
for the purposes of this calculation.
Sample sizes with regard to binary data:
n = [t2 N p(1-p)] / [t2 p(1-p) + 2 (N-1)]
with N being the size of the total number of cases, n being the sample
size,  the expected error, t being the value taken from the t
distribution corresponding to a certain confidence interval, and p being
the probability of an event.
For a finite population of size N, the standard error of the sample mean
of size n, is:
[(N -n)/(nN)]½
There are several formulas for the sample size needed for a t-test. The
simplest one is
n = 2(Z+Z)22/D2
which underestimates the sample size, but is reasonable for large
sample sizes. A less inaccurate formula replaces the Z values with t
values, and requires iteration, since the df for the t distribution
depends on the sample size. The accurate formula uses a non-central t
distribution and it also requires iteration.
The simplest approximation is to replace the first Z value in the above
formula with the value from the studentized range statistic that is used
to derive Tukey's follow-up test. If you don't have sufficiently detailed
tables of the studentized range, you can approximate the Tukey
follow-up test using a Bonferroni correction. That is, change the first Z
value to Z where k is the number of comparisons.
Neither of these solutions is exact and the exact solution is a bit messy.
But either of the above approaches is probably close enough,
especially if the resulting sample size is larger than (say) 30.
A better stopping rule for conventional statistical tests is as follows:
Test some minimum (pre-determined) number of subjects.
Stop if p-value is equal to or less than .01, or p-value equal to or
greater than .36; otherwise, run more subjects.
Obviously, another option is to stop if/when the number of subjects
becomes too great for the effect to be of practical interest. This
procedure maintains  about 0.05.
We may categorized probability proportion to size (PPS) sampling,
stratification, and ratio estimation (or any other form of model assisted
estimation) as tools that protect one from the results of a very unlucky
sample. The first two (PPS sampling and stratification) do this by
manipulation of the sampling plan (with PPS sampling conceptually a
limiting case of stratification). Model assisted estimation methods
such as ratio estimation serve the same purpose by introduction of
ancillary information into the estimation procedure. Which tools are
preferable depends, as others have said, on costs, availability of
information that allows use of these tools, and the potential payoffs
(none of these will help much if the stratification/PPS/ratio estimation
variable is not well correlated with the response variable of interest).
There are also heuristic methods for determination of sample size. For
example, in healthcare behavior and process measurement sampling
criteria are designed for a 95% CI of 10 percentage points around a
population mean of 0.50; There is a heuristic rule: "If the number of
individuals in the target population is smaller than 50 per month,
systems do not use sampling procedures but, attempt to collect data
from all individuals in the target population."