Point and interval estimation

Estimation is the process of making inferences from a sample about an unknown population parameter. An estimator is a statistic that is used to infer the value of an unknown parameter.

A point estimate is the best estimate, in some sense, of the parameter based on a sample. It should be obvious that any point estimate is not absolutely accurate. It is an estimate based on only a single random sample. If repeated random samples were taken from the population, the point estimate would be expected to vary from sample to sample.

A confidence interval is an estimate constructed on the basis that a specified proportion of the confidence intervals include the true parameter in repeated sampling. How frequently the confidence interval contains the parameter is determined by the confidence level. 95% is commonly used and means that in repeated sampling 95% of the confidence intervals include the parameter. 99% is sometimes used when more confidence is needed and means that in repeated sampling 99% of the intervals include the true parameter. It is unusual to use a confidence level of less than 90% as too many intervals would fail to include the parameter. Likewise, confidence levels larger than 99% are not used often because the intervals become wider the higher the confidence level and therefore require large sample sizes to make usable intervals.

Many people misunderstand confidence intervals. A confidence interval does not predict with a given probability that the parameter lies within the interval. The problem arises because the word confidence is misinterpreted as implying probability. In frequentist statistics, probability statements cannot be made about parameters. Parameters are fixed, not random variables, and so a probability statement cannot be made about them. When a confidence interval has been constructed, it either does or does not include the parameter.

In recent years the use of confidence intervals has become more common. A confidence interval provides much more information than just a hypothesis test p-value. It indicates the uncertainty of an estimate of a parameter and allows you to consider the practical importance, rather than just statistical significance.

Confidence intervals and hypothesis tests are closely related. Most introductory textbooks discuss how confidence intervals are equivalent to a hypothesis test. When the 95% confidence interval contains the hypothesized value the hypothesis test is statistically significant at the 5% significance level, and when it does not contain the value the test is not significant. Likewise, with a 99% confidence interval and hypothesis test at the 1% significance level. A confidence interval can be considered as the set of parameter values consistent with the data at some specified level, as assessed by testing each possible value in turn.

Th relationship between hypothesis tests and confidence interval only holds when the estimator and the hypothesis test both use the same underlying evidence function. When a software package uses different evidence functions for the confidence interval and hypothesis test, the results can be inconsistent. The hypothesis test may be statistically significant, but the confidence interval may include the hypothesized value suggesting the result is not significant. Where possible the same underlying evidence function should be used to form the confidence interval and test the hypotheses. Be aware that not many statistical software packages follow this rule!

The following example illustrates the use of a point estimate, confidence interval and hypothesis test in making inferences.

A scientist might study the difference in blood cholesterol between a new drug treatment and a placebo. Improvements in cholesterol greater than 20mg/dL would be considered practically important, and lead to a change in the treatment of patients, but smaller differences would not. The possible outcomes of the study in terms of a point-estimate, confidence interval estimate, and hypothesis test might be:

Point estimate 95% confidence interval Hypothesis test (5% significance level) Interpretation
0 mg/dL -5 to 5 mg/dL Not significant The confidence interval is within the range of no practical difference, 20mg/dL.

There is clear evidence the treatment does not produce a difference of practical importance.

10 mg/dL -5 to 25 mg/dL Not significant Part of the confidence interval lies outside the range of no practical difference, 20mg/dL.

Although the hypothesis test is not significant, there may be an important practical difference, though a larger sample size is required to make any sharper inferences.

10 mg/dL 5 to 15 mg/dL Significant The confidence interval excludes the null (zero) difference but still lies within the range of no practical difference, 20mg/dL.

Although the hypothesis test is statistically significant, the difference is of no practical importance.

30 mg/dL 25 to 45 mg/dL Significant The confidence interval excludes the null difference and lies outside the range of no practical difference, 20mg/dL.

The hypothesis test is statistically significant, and the difference is of practical importance.

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