In statistics, a confidence interval (CI) is a type of interval
estimate (of a population parameter) that is computed from the
observed data. The confidence level is the frequency (i.e., the
proportion) of possible confidence intervals that contain the true
value of their corresponding parameter. In other words, if confidence
intervals are constructed using a given confidence level in an
infinite number of independent experiments, the proportion of those
intervals that contain the true value of the parameter will match the
confidence level.[1][2][3]
Confidence intervals consist of a range of values (interval) that act
as good estimates of the unknown population parameter. However, the
interval computed from a particular sample does not necessarily
include the true value of the parameter. Since the observed data are
random samples from the true population, the confidence interval
obtained from the data is also random. If a corresponding hypothesis
test is performed, the confidence level is the complement of the level
of significance; for example, a 95% confidence interval reflects a
significance level of 0.05.[4] If it is hypothesized that a true
parameter value is 0 but the 95% confidence interval does not contain
0, then the estimate is significantly different from zero at the 5%
significance level.
The desired level of confidence is set by the researcher (not
determined by data). Most commonly, the 95% confidence level is
used.[5] However, other confidence levels can be used, for example,
90% and 99%.
Factors affecting the width of the confidence interval include the
size of the sample, the confidence level, and the variability in the
sample. A larger sample size normally will lead to a better estimate
of the population parameter.
Confidence intervals were introduced to statistics by
Contents 1 Conceptual basis 1.1 Introduction 1.2 Meaning and interpretation 1.2.1 Misunderstandings 1.3 Philosophical issues 1.4 Relationship with other statistical topics 1.4.1 Statistical hypothesis testing 1.4.2 Confidence region 1.4.3 Confidence band 2 Basic steps 3 Statistical theory 3.1 Definition 3.1.1 Approximate confidence intervals 3.2 Desirable properties 3.3 Methods of derivation 4 Examples 4.1 Practical example 4.1.1 Interpretation 4.2 Theoretical example 5 Alternatives and critiques 5.1 Comparison to prediction intervals 5.2 Comparison to tolerance intervals 5.3 Comparison to Bayesian interval estimates 5.4 Confidence intervals for proportions and related quantities 5.5 Counter-examples 5.5.1 Confidence procedure for uniform location 5.5.2 Confidence procedure for ω2 6 See also 6.1
7 References 8 Bibliography 8.1 External links 8.2 Online calculators Conceptual basis[edit] In this bar chart, the top ends of the brown bars indicate observed means and the red line segments ("error bars") represent the confidence intervals around them. Although the error bars are shown as symmetric around the means, that is not always the case. It is also important that in most graphs, the error bars do not represent confidence intervals (e.g., they often represent standard errors or standard deviations) Introduction[edit] Interval estimates can be contrasted with point estimates. A point estimate is a single value given as the estimate of a population parameter that is of interest, for example, the mean of some quantity. An interval estimate specifies instead a range within which the parameter is estimated to lie. Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates. For example, a confidence interval can be used to describe how reliable survey results are. In a poll of election–voting intentions, the result might be that 40% of respondents intend to vote for a certain party. A 99% confidence interval for the proportion in the whole population having the same intention on the survey might be 30% to 50%. From the same data one may calculate a 90% confidence interval, which in this case might be 37% to 43%. A major factor determining the length of a confidence interval is the size of the sample used in the estimation procedure, for example, the number of people taking part in a survey. Meaning and interpretation[edit] See also: § Practical Example Interpretation Various interpretations of a confidence interval can be given (taking the 90% confidence interval as an example in the following). The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on numerous samples, the fraction of calculated confidence intervals (which would differ for each sample) that encompass the true population parameter would tend toward 90%."[1] The confidence interval can be expressed in terms of a single sample: "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." Note this is a probability statement about the confidence interval, not the population parameter. This considers the probability associated with a confidence interval from a pre-experiment point of view, in the same context in which arguments for the random allocation of treatments to study items are made. Here the experimenter sets out the way in which they intend to calculate a confidence interval and to know, before they do the actual experiment, that the interval they will end up calculating has a particular chance of covering the true but unknown value.[3] This is very similar to the "repeated sample" interpretation above, except that it avoids relying on considering hypothetical repeats of a sampling procedure that may not be repeatable in any meaningful sense. See Neyman construction. The explanation of a confidence interval can amount to something like: "The confidence interval represents values for the population parameter for which the difference between the parameter and the observed estimate is not statistically significant at the 10% level".[6] In fact, this relates to one particular way in which a confidence interval may be constructed. In each of the above, the following applies: If the true value of the parameter lies outside the 90% confidence interval, then a sampling event has occurred (namely, obtaining a point estimate of the parameter at least this far from the true parameter value) which had a probability of 10% (or less) of happening by chance. Misunderstandings[edit] See also: § Counter-examples See also: Misunderstandings of p-values Confidence intervals are frequently misunderstood, and published studies have shown that even professional scientists often misinterpret them.[7][8][9][10] A 95% confidence interval does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).[11] Once an experiment is done and an interval calculated, this interval either covers the parameter value or it does not; it is no longer a matter of probability. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval.[12] Neyman himself (the original proponent of confidence intervals) made this point in his original paper:[3] "It will be noticed that in the above description, the probability statements refer to the problems of estimation with which the statistician will be concerned in the future. In fact, I have repeatedly stated that the frequency of correct results will tend to α. Consider now the case when a sample is already drawn, and the calculations have given [particular limits]. Can we say that in this particular case the probability of the true value [falling between these limits] is equal to α? The answer is obviously in the negative. The parameter is an unknown constant, and no probability statement concerning its value may be made..." Deborah Mayo expands on this further as follows:[13] "It must be stressed, however, that having seen the value [of the data], Neyman-Pearson theory never permits one to conclude that the specific confidence interval formed covers the true value of 0 with either (1 − α)100% probability or (1 − α)100% degree of confidence. Seidenfeld's remark seems rooted in a (not uncommon) desire for Neyman-Pearson confidence intervals to provide something which they cannot legitimately provide; namely, a measure of the degree of probability, belief, or support that an unknown parameter value lies in a specific interval. Following Savage (1962), the probability that a parameter lies in a specific interval may be referred to as a measure of final precision. While a measure of final precision may seem desirable, and while confidence levels are often (wrongly) interpreted as providing such a measure, no such interpretation is warranted. Admittedly, such a misinterpretation is encouraged by the word 'confidence'." A 95% confidence interval does not mean that 95% of the sample data lie within the interval. A confidence interval is not a definitive range of plausible values for the sample parameter, though it may be understood as an estimate of plausible values for the population parameter. A particular confidence interval of 95% calculated from an experiment does not mean that there is a 95% probability of a sample parameter from a repeat of the experiment falling within this interval. Philosophical issues[edit]
The principle behind confidence intervals was formulated to provide an
answer to the question raised in statistical inference of how to deal
with the uncertainty inherent in results derived from data that are
themselves only a randomly selected subset of a population. There are
other answers, notably that provided by
1. Identify the sample mean, x ¯ displaystyle bar x . 2. Identify whether the standard deviation is known, σ displaystyle sigma , or unknown, s. If standard deviation is known then z ∗ = Φ − 1 ( α 2 ) = 1 − Φ − 1 ( 1 − α 2 ) displaystyle z^ * =Phi ^ -1 left( frac alpha 2 right)=1-Phi ^ -1 left(1- frac alpha 2 right) , where Φ displaystyle Phi is the C.D.F. of the Gaussian distribution, is used as the critical value. This value is only dependent on the confidence level for the test. Typical two sided confidence levels are:[23] C z* 99% 2.576 98% 2.326 95% 1.96 90% 1.645 If the standard deviation is unknown then Student's t distribution is used as the critical value. This value is dependent on the confidence level (C) for the test and degrees of freedom. The degrees of freedom is found by subtracting one from the number of observations, n − 1. The critical value is found from the t-distribution table. In this table the critical value is written as tα(r), where r is the degrees of freedom and α = 1 − C 2 displaystyle alpha = 1-C over 2 . 3. Plug the found values into the appropriate equations: For a known standard deviation: ( x ¯ − z ∗ σ n , x ¯ + z ∗ σ n ) displaystyle left( bar x -z^ * sigma over sqrt n , bar x +z^ * sigma over sqrt n right) For an unknown standard deviation: ( x ¯ − t ∗ s n , x ¯ + t ∗ s n ) displaystyle left( bar x -t^ * s over sqrt n , bar x +t^ * s over sqrt n right) 4. The final step is to interpret the answer. Since the found answer is an interval with an upper and lower bound it is appropriate to state that based on the given data we are __ % (dependent on the confidence level) confident that the true mean of the population is between __ (lower bound) and __ (upper bound).[24] Statistical theory[edit] Definition[edit] Let X be a random sample from a probability distribution with statistical parameters θ, which is a quantity to be estimated, and φ, representing quantities that are not of immediate interest. A confidence interval for the parameter θ, with confidence level or confidence coefficient γ, is an interval with random endpoints (u(X), v(X)), determined by the pair of random variables u(X) and v(X), with the property: Pr θ , φ ( u ( X ) < θ < v ( X ) ) = γ for all ( θ , φ ) . displaystyle Pr _ theta ,varphi (u(X)<theta <v(X))=gamma text for all (theta ,varphi ). The quantities φ in which there is no immediate interest are called nuisance parameters, as statistical theory still needs to find some way to deal with them. The number γ, with typical values close to but not greater than 1, is sometimes given in the form 1 − α (or as a percentage 100%·(1 − α)), where α is a small non-negative number, close to 0. Here Prθ,φ indicates the probability distribution of X characterised by (θ, φ). An important part of this specification is that the random interval (u(X), v(X)) covers the unknown value θ with a high probability no matter what the true value of θ actually is. Note that here Prθ,φ need not refer to an explicitly given parameterized family of distributions, although it often does. Just as the random variable X notionally corresponds to other possible realizations of x from the same population or from the same version of reality, the parameters (θ, φ) indicate that we need to consider other versions of reality in which the distribution of X might have different characteristics. In a specific situation, when x is the outcome of the sample X, the interval (u(x), v(x)) is also referred to as a confidence interval for θ. Note that it is no longer possible to say that the (observed) interval (u(x), v(x)) has probability γ to contain the parameter θ. This observed interval is just one realization of all possible intervals for which the probability statement holds. Approximate confidence intervals[edit] In many applications, confidence intervals that have exactly the required confidence level are hard to construct. But practically useful intervals can still be found: the rule for constructing the interval may be accepted as providing a confidence interval at level γ if Pr θ , φ ( u ( X ) < θ < v ( X ) ) ≈ γ for all ( θ , φ ) displaystyle Pr _ theta ,varphi (u(X)<theta <v(X))approx gamma text for all (theta ,varphi ), to an acceptable level of approximation. Alternatively, some authors[25] simply require that Pr θ , φ ( u ( X ) < θ < v ( X ) ) ≥ γ for all ( θ , φ ) displaystyle Pr _ theta ,varphi (u(X)<theta <v(X))geq gamma text for all (theta ,varphi ), which is useful if the probabilities are only partially identified, or imprecise. Desirable properties[edit] When applying standard statistical procedures, there will often be standard ways of constructing confidence intervals. These will have been devised so as to meet certain desirable properties, which will hold given that the assumptions on which the procedure rely are true. These desirable properties may be described as: validity, optimality, and invariance. Of these "validity" is most important, followed closely by "optimality". "Invariance" may be considered as a property of the method of derivation of a confidence interval rather than of the rule for constructing the interval. In non-standard applications, the same desirable properties would be sought. Validity. This means that the nominal coverage probability (confidence level) of the confidence interval should hold, either exactly or to a good approximation. Optimality. This means that the rule for constructing the confidence interval should make as much use of the information in the data-set as possible. Recall that one could throw away half of a dataset and still be able to derive a valid confidence interval. One way of assessing optimality is by the length of the interval so that a rule for constructing a confidence interval is judged better than another if it leads to intervals whose lengths are typically shorter. Invariance. In many applications, the quantity being estimated might not be tightly defined as such. For example, a survey might result in an estimate of the median income in a population, but it might equally be considered as providing an estimate of the logarithm of the median income, given that this is a common scale for presenting graphical results. It would be desirable that the method used for constructing a confidence interval for the median income would give equivalent results when applied to constructing a confidence interval for the logarithm of the median income: specifically the values at the ends of the latter interval would be the logarithms of the values at the ends of former interval. Methods of derivation[edit] For non-standard applications, there are several routes that might be taken to derive a rule for the construction of confidence intervals. Established rules for standard procedures might be justified or explained via several of these routes. Typically a rule for constructing confidence intervals is closely tied to a particular way of finding a point estimate of the quantity being considered. Summary statistics This is closely related to the method of moments for estimation. A simple example arises where the quantity to be estimated is the mean, in which case a natural estimate is the sample mean. The usual arguments indicate that the sample variance can be used to estimate the variance of the sample mean. A confidence interval for the true mean can be constructed centered on the sample mean with a width which is a multiple of the square root of the sample variance. Likelihood theory Where estimates are constructed using the maximum likelihood principle, the theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates.[clarification needed] One way is by using Wilks's theorem to find all the possible values of θ displaystyle theta that fulfill the following restriction:[26] ln ( L ( θ ) ) ≥ ln ( L ( θ ^ ) ) − 1 2 χ 1 , 1 − α 2 displaystyle ln(L(theta ))geq ln(L( hat theta ))- frac 1 2 chi _ 1,1-alpha ^ 2 Estimating equations
The estimation approach here can be considered as both a
generalization of the method of moments and a generalization of the
maximum likelihood approach. There are corresponding generalizations
of the results of maximum likelihood theory that allow confidence
intervals to be constructed based on estimates derived from estimating
equations.[clarification needed]
Examples[edit] Practical example[edit] A machine fills cups with a liquid, and is supposed to be adjusted so that the content of the cups is 250 g of liquid. As the machine cannot fill every cup with exactly 250.0 g, the content added to individual cups shows some variation, and is considered a random variable X. This variation is assumed to be normally distributed around the desired average of 250 g, with a standard deviation, σ, of 2.5 g. To determine if the machine is adequately calibrated, a sample of n = 25 cups of liquid is chosen at random and the cups are weighed. The resulting measured masses of liquid are X1, ..., X25, a random sample from X. To get an impression of the expectation μ, it is sufficient to give an estimate. The appropriate estimator is the sample mean: μ ^ = X ¯ = 1 n ∑ i = 1 n X i . displaystyle hat mu = bar X = frac 1 n sum _ i=1 ^ n X_ i . The sample shows actual weights x1, ..., x25, with mean: x ¯ = 1 25 ∑ i = 1 25 x i = 250.2 grams . displaystyle bar x = frac 1 25 sum _ i=1 ^ 25 x_ i =250.2 text grams . If we take another sample of 25 cups, we could easily expect to find mean values like 250.4 or 251.1 grams. A sample mean value of 280 grams however would be extremely rare if the mean content of the cups is in fact close to 250 grams. There is a whole interval around the observed value 250.2 grams of the sample mean within which, if the whole population mean actually takes a value in this range, the observed data would not be considered particularly unusual. Such an interval is called a confidence interval for the parameter μ. How do we calculate such an interval? The endpoints of the interval have to be calculated from the sample, so they are statistics, functions of the sample X1, ..., X25 and hence random variables themselves. In our case we may determine the endpoints by considering that the sample mean X from a normally distributed sample is also normally distributed, with the same expectation μ, but with a standard error of: σ n = 2.5 g 25 = 0.5 grams displaystyle frac sigma sqrt n = frac 2.5 text g sqrt 25 =0.5 text grams By standardizing, we get a random variable: Z = X ¯ − μ σ / n = X ¯ − μ 0.5 displaystyle Z= frac bar X -mu sigma / sqrt n = frac bar X -mu 0.5 dependent on the parameter μ to be estimated, but with a standard normal distribution independent of the parameter μ. Hence it is possible to find numbers −z and z, independent of μ, between which Z lies with probability 1 − α, a measure of how confident we want to be. We take 1 − α = 0.95, for example. So we have: P ( − z ≤ Z ≤ z ) = 1 − α = 0.95. displaystyle P(-zleq Zleq z)=1-alpha =0.95. The number z follows from the cumulative distribution function, in this case the cumulative normal distribution function: Φ ( z ) = P ( Z ≤ z ) = 1 − α 2 = 0.975 , z = Φ − 1 ( Φ ( z ) ) = Φ − 1 ( 0.975 ) = 1.96 , displaystyle begin aligned Phi (z)&=P(Zleq z)=1- tfrac alpha 2 =0.975,\[6pt]z&=Phi ^ -1 (Phi (z))=Phi ^ -1 (0.975)=1.96,end aligned and we get: 0.95 = 1 − α = P ( − z ≤ Z ≤ z ) = P ( − 1.96 ≤ X ¯ − μ σ / n ≤ 1.96 ) = P ( X ¯ − 1.96 σ n ≤ μ ≤ X ¯ + 1.96 σ n ) . displaystyle begin aligned 0.95&=1-alpha =P(-zleq Zleq z)=Pleft(-1.96leq frac bar X -mu sigma / sqrt n leq 1.96right)\[6pt]&=Pleft( bar X -1.96 frac sigma sqrt n leq mu leq bar X +1.96 frac sigma sqrt n right).end aligned In other words, the lower endpoint of the 95% confidence interval is: Lower endpoint = X ¯ − 1.96 σ n , displaystyle text Lower endpoint = bar X -1.96 frac sigma sqrt n , and the upper endpoint of the 95% confidence interval is: Upper endpoint = X ¯ + 1.96 σ n . displaystyle text Upper endpoint = bar X +1.96 frac sigma sqrt n . With the values in this example, the confidence interval is: 0.95 = Pr ( X ¯ − 1.96 × 0.5 ≤ μ ≤ X ¯ + 1.96 × 0.5 ) = Pr ( X ¯ − 0.98 ≤ μ ≤ X ¯ + 0.98 ) . displaystyle begin aligned 0.95&=Pr( bar X -1.96times 0.5leq mu leq bar X +1.96times 0.5)\[6pt]&=Pr( bar X -0.98leq mu leq bar X +0.98).end aligned As the standard deviation of the population σ is known in this case, the distribution of the sample mean X ¯ displaystyle bar X is a normal distribution with μ displaystyle mu the only unknown parameter. In the theoretical example below, the parameter σ is also unknown, which calls for using the Student's t-distribution. Interpretation[edit] This might be interpreted as: with probability 0.95 we will find a confidence interval in which the value of parameter μ will be between the stochastic endpoints X ¯ − 0 . 98 displaystyle ! bar X -0 . 98 and X ¯ + 0.98. displaystyle ! bar X +0.98. This does not mean there is 0.95 probability that the value of parameter μ is in the interval obtained by using the currently computed value of the sample mean, ( x ¯ − 0.98 , x ¯ + 0.98 ) . displaystyle ( bar x -0.98,, bar x +0.98). Instead, every time the measurements are repeated, there will be another value for the mean X of the sample. In 95% of the cases μ will be between the endpoints calculated from this mean, but in 5% of the cases it will not be. The actual confidence interval is calculated by entering the measured masses in the formula. Our 0.95 confidence interval becomes: ( x ¯ − 0.98 ; x ¯ + 0.98 ) = ( 250.2 − 0.98 ; 250.2 + 0.98 ) = ( 249.22 ; 251.18 ) . displaystyle ( bar x -0.98; bar x +0.98)=(250.2-0.98;250.2+0.98)=(249.22;251.18)., The blue vertical line segments represent 50 realizations of a confidence interval for the population mean μ, represented as a red horizontal dashed line; note that some confidence intervals do not contain the population mean, as expected. In other words, the 95% confidence interval is between the lower endpoint 249.22 g and the upper endpoint 251.18 g. As the desired value 250 of μ is within the resulted confidence interval, there is no reason to believe the machine is wrongly calibrated. The calculated interval has fixed endpoints, where μ might be in between (or not). Thus this event has probability either 0 or 1. One cannot say: "with probability (1 − α) the parameter μ lies in the confidence interval." One only knows that by repetition in 100(1 − α) % of the cases, μ will be in the calculated interval. In 100α% of the cases however it does not. And unfortunately one does not know in which of the cases this happens. That is (instead of using the term "probability") why one can say: "with confidence level 100(1 − α) %, μ lies in the confidence interval." The maximum error is calculated to be 0.98 since it is the difference between the value that we are confident of with upper or lower endpoint. The figure on the right shows 50 realizations of a confidence interval for a given population mean μ. If we randomly choose one realization, the probability is 95% we end up having chosen an interval that contains the parameter; however, we may be unlucky and have picked the wrong one. We will never know; we are stuck with our interval. Theoretical example[edit] Suppose X1, ..., Xn is an independent sample from a normally distributed population with unknown (parameters) mean μ and variance σ2. Let X ¯ = ( X 1 + ⋯ + X n ) / n , displaystyle bar X =(X_ 1 +cdots +X_ n )/n,, S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ¯ ) 2 . displaystyle S^ 2 = frac 1 n-1 sum _ i=1 ^ n (X_ i - bar X ,)^ 2 . Where X is the sample mean, and S2 is the sample variance. Then T = X ¯ − μ S / n displaystyle T= frac bar X -mu S/ sqrt n has a
Pr ( − c ≤ T ≤ c ) = 0.95 displaystyle Pr(-cleq Tleq c)=0.95, ("97.5th" and "0.95" are correct in the preceding expressions. There is a 2.5% chance that T will be less than −c and a 2.5% chance that it will be larger than +c. Thus, the probability that T will be between −c and +c is 95%.) Consequently, Pr ( X ¯ − c S n ≤ μ ≤ X ¯ + c S n ) = 0.95 displaystyle Pr left( bar X - frac cS sqrt n leq mu leq bar X + frac cS sqrt n right)=0.95, and we have a theoretical (stochastic) 95% confidence interval for μ. After observing the sample we find values x for X and s for S, from which we compute the confidence interval [ x ¯ − c s n , x ¯ + c s n ] , displaystyle left[ bar x - frac cs sqrt n , bar x + frac cs sqrt n right],, an interval with fixed numbers as endpoints, of which we can no longer
say there is a certain probability it contains the parameter μ;
either μ is in this interval or isn't.
Alternatives and critiques[edit]
Main article: Interval estimation
Confidence intervals are one method of interval estimation, and the
most widely used in frequentist statistics. An analogous concept in
Pr θ , φ ( u ( X ) < Y < v ( X ) ) = γ for all ( θ , φ ) . displaystyle Pr _ theta ,varphi (u(X)<Y<v(X))=gamma text for all (theta ,varphi )., Here Prθ,φ indicates the joint probability distribution of the random variables (X, Y), where this distribution depends on the statistical parameters (θ, φ). Comparison to tolerance intervals[edit] Main article: Tolerance interval This section needs expansion. You can help by adding to it. (September 2014) Comparison to Bayesian interval estimates[edit]
See also:
Pr ( u ( x ) < Θ < v ( x ) ∣ X = x ) = γ . displaystyle Pr(u(x)<Theta <v(x)mid X=x)=gamma ., Here Θ is used to emphasize that the unknown value of θ is being treated as a random variable. The definitions of the two types of intervals may be compared as follows. The definition of a confidence interval involves probabilities calculated from the distribution of X for a given (θ, φ) (or conditional on these values) and the condition needs to hold for all values of (θ, φ). The definition of a credible interval involves probabilities calculated from the distribution of Θ conditional on the observed values of X = x and marginalised (or averaged) over the values of Φ, where this last quantity is the random variable corresponding to the uncertainty about the nuisance parameters in φ. Note that the treatment of the nuisance parameters above is often
omitted from discussions comparing confidence and credible intervals
but it is markedly different between the two cases.
In some simple standard cases, the intervals produced as confidence
and credible intervals from the same data set can be identical. They
are very different if informative prior information is included in the
Bayesian analysis, and may be very different for some parts of the
space of possible data even if the Bayesian prior is relatively
uninformative.
There is disagreement about which of these methods produces the most
useful results: the mathematics of the computations are rarely in
question–confidence intervals being based on sampling distributions,
credible intervals being based on Bayes' theorem–but the application
of these methods, the utility and interpretation of the produced
statistics, is debated.
Confidence intervals for proportions and related quantities[edit]
See also:
X 1 , X 2 displaystyle X_ 1 ,X_ 2 are independent observations from a Uniform(θ − 1/2, θ + 1/2) distribution. Then the optimal 50% confidence procedure[31] is X ¯ ±
X 1 − X 2
2 if
X 1 − X 2
< 1 / 2 1 −
X 1 − X 2
2 if
X 1 − X 2
≥ 1 / 2. displaystyle bar X pm begin cases dfrac X_ 1 -X_ 2 2 & text if X_ 1 -X_ 2 <1/2\[8pt] dfrac 1-X_ 1 -X_ 2 2 & text if X_ 1 -X_ 2 geq 1/2.end cases A fiducial or objective Bayesian argument can be used to derive the interval estimate X ¯ ± 1 −
X 1 − X 2
4 , displaystyle bar X pm frac 1-X_ 1 -X_ 2 4 , which is also a 50% confidence procedure. Welch showed that the first confidence procedure dominates the second, according to desiderata from confidence interval theory; for every θ 1 ≠ θ displaystyle theta _ 1 neq theta , the probability that the first procedure contains θ 1 displaystyle theta _ 1 is less than or equal to the probability that the second procedure contains θ 1 displaystyle theta _ 1 . The average width of the intervals from the first procedure is less than that of the second. Hence, the first procedure is preferred under classical confidence interval theory. However, when
X 1 − X 2
≥ 1 / 2 displaystyle X_ 1 -X_ 2 geq 1/2 , intervals from the first procedure are guaranteed to contain the true value θ displaystyle theta : Therefore, the nominal 50% confidence coefficient is unrelated to the uncertainty we should have that a specific interval contains the true value. The second procedure does not have this property. Moreover, when the first procedure generates a very short interval, this indicates that X 1 , X 2 displaystyle X_ 1 ,X_ 2 are very close together and hence only offer the information in a single data point. Yet the first interval will exclude almost all reasonable values of the parameter due to its short width. The second procedure does not have this property. The two counter-intuitive properties of the first procedure — 100% coverage when X 1 , X 2 displaystyle X_ 1 ,X_ 2 are far apart and almost 0% coverage when X 1 , X 2 displaystyle X_ 1 ,X_ 2 are close together — balance out to yield 50% coverage on average. However, despite the first procedure being optimal, its intervals offer neither an assessment of the precision of the estimate nor an assessment of the uncertainty one should have that the interval contains the true value. This counter-example is used to argue against naïve interpretations of confidence intervals. If a confidence procedure is asserted to have properties beyond that of the nominal coverage (such as relation to precision, or a relationship with Bayesian inference), those properties must be proved; they do not follow from the fact that a procedure is a confidence procedure. Confidence procedure for ω2[edit] Steiger[32] suggested a number of confidence procedures for common effect size measures in ANOVA. Morey et al.[11] point out that several of these confidence procedures, including the one for ω2, have the property that as the F statistic becomes increasingly small — indicating misfit with all possible values of ω2 — the confidence interval shrinks and can even contain only the single value ω2=0; that is, the CI is infinitesimally narrow (this occurs when p ≥ 1 − α / 2 displaystyle pgeq 1-alpha /2 for a 100 ( 1 − α ) % displaystyle 100(1-alpha )% CI). This behavior is consistent with the relationship between the confidence procedure and significance testing: as F becomes so small that the group means are much closer together than we would expect by chance, a significance test might indicate rejection for most or all values of ω2. Hence the interval will be very narrow or even empty (or, by a convention suggested by Steiger, containing only 0). However, this does not indicate that the estimate of ω2 is very precise. In a sense, it indicates the opposite: that the trustworthiness of the results themselves may be in doubt. This is contrary to the common interpretation of confidence intervals that they reveal the precision of the estimate. See also[edit] Cumulative distribution function-based nonparametric confidence interval CLs upper limits (particle physics) Confidence distribution Credence (statistics) Error bar Estimation statistics p-value Robust confidence intervals Confidence region
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& Hall, p49, p209
^ Kendall, M.G. and Stuart, D.G. (1973) The Advanced Theory of
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Section 20.4
^ a b c d Neyman, J. (1937). "Outline of a Theory of Statistical
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^ Cox D.R., Hinkley D.V. (1974) Theoretical Statistics, Chapman &
Hall, p214, 225, 233
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^ [2]
^ Hoekstra, R., R. D. Morey, J. N. Rouder, and E-J. Wagenmakers, 2014.
Robust misinterpretation of confidence intervals. Psychonomic Bulletin
Review, in press. [3]
^ Scientists’ grasp of confidence intervals doesn’t inspire
confidence, Science News, July 3, 2014
^ a b Morey, R. D.; Hoekstra, R.; Rouder, J. N.; Lee, M. D.;
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Confidence Intervals". Psychonomic Bulletin & Review. 23 (1):
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^ "1.3.5.2. Confidence Limits for the Mean". nist.gov.
^ Mayo, D. G. (1981) "In defence of the Neyman-Pearson theory of
confidence intervals", Philosophy of Science, 48 (2), 269–280.
JSTOR 187185
^ T. Seidenfeld, Philosophical Problems of Statistical Inference:
Learning from R.A. Fisher, Springer-Verlag, 1979
^ "
Bibliography[edit] Fisher, R.A. (1956) Statistical Methods and Scientific Inference.
Oliver and Boyd, Edinburgh. (See p. 32.)
Freund, J.E. (1962) Mathematical
Mehta, S. (2014)
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