View in EDS HTML Full Text PDF Full Text

Introduction to Sample Size Choice for Confidence Intervals Based on 't' Statistics

Saved in:
Bibliographic Details
Title: Introduction to Sample Size Choice for Confidence Intervals Based on 't' Statistics
Language: English
Authors: Liu, Xiaofeng Steven, Loudermilk, Brandon, Simpson, Thomas
Source: Measurement in Physical Education and Exercise Science. 2014 18(2):91-100.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 10
Publication Date: 2014
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Sample Size, Statistical Analysis, Confidence Testing, Intervals, Measurement Techniques, Computation, Exercise
DOI: 10.1080/1091367X.2013.864657
ISSN: 1091-367X
Abstract: Sample size can be chosen to achieve a specified width in a confidence interval. The probability of obtaining a narrow width given that the confidence interval includes the population parameter is defined as the power of the confidence interval, a concept unfamiliar to many practitioners. This article shows how to utilize the Statistical Analysis System (SAS) proc power procedure to determine an appropriate sample size and achieve sufficient power for a specified confidence interval. Two examples in sport and exercise science are used to illustrate sample size determination for confidence intervals in the dependent and independent "t" tests. The relevant SAS code and output are provided with detailed annotations. As the use of confidence intervals becomes a more integral part of studies in sport and exercise science, the reporting and analysis of their power should be considered much like that of hypothesis tests.
Abstractor: As Provided
Number of References: 18
Entry Date: 2014
Accession Number: EJ1029741
Database: ERIC
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
    Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwG70oR4LbuA-lnJZlh33Q8gAAAA4jCB3wYJKoZIhvcNAQcGoIHRMIHOAgEAMIHIBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDOwGJRCUjdcLR5PiqQIBEICBmm3SO8pP5lbgFUoM4WNTldAXp_vbwHpWcxLRkk5T3ocblxfTpOBy5U94T3e_ObKHiXimqS3diELSiaNayNs94ffdmdtD2Na27_jCLnqoFDVJcEYdkwiVbfgZmzI4F66d-d6H7HxQw8gDgyitnrsnQEjhP1-WONUxGBWZo5m9rmmHzpXB4O3dCPwDbEVAAlJe5JB3eDE85l1AIvU=
Text:
  Availability: 1
  Value: <anid>AN0094643633;7mm01apr.14;2019Mar26.12:55;v2.2.500</anid> <title id="AN0094643633-1">Introduction to Sample Size Choice for Confidence Intervals Based on t Statistics. </title> <p>Sample size can be chosen to achieve a specified width in a confidence interval. The probability of obtaining a narrow width given that the confidence interval includes the population parameter is defined as the power of the confidence interval, a concept unfamiliar to many practitioners. This article shows how to utilize the Statistical Analysis System (SAS) proc power procedure to determine an appropriate sample size and achieve sufficient power for a specified confidence interval. Two examples in sport and exercise science are used to illustrate sample size determination for confidence intervals in the dependent and independent t tests. The relevant SAS code and output are provided with detailed annotations. As the use of confidence intervals becomes a more integral part of studies in sport and exercise science, the reporting and analysis of their power should be considered much like that of hypothesis tests.</p> <p>Keywords: confidence interval; SAS; sample size</p> <p>Confidence intervals have received great emphasis in the reporting of scientific findings. A few journals have issued editorial policies that encourage people to use confidence intervals instead of hypothesis tests ([<reflink idref="bib10" id="ref1">10</reflink>]; [<reflink idref="bib18" id="ref2">18</reflink>]). Null hypothesis testing is usually used to rule out chance as a plausible explanation of the observed treatment effect in a study. Almost all the studies in the social sciences culminate in a ritualistic <emph>p</emph>-value followed by declaration of a yes-or-no answer to the original research question. Although null hypothesis testing has long been established, it has recently become the target of frequent criticism in the social sciences.</p> <p>The critics of null hypothesis testing argued that its inverted logic is counterintuitive to statistical consumers ([<reflink idref="bib2" id="ref3">2</reflink>]; [<reflink idref="bib5" id="ref4">5</reflink>]; [<reflink idref="bib6" id="ref5">6</reflink>]; [<reflink idref="bib9" id="ref6">9</reflink>]; [<reflink idref="bib16" id="ref7">16</reflink>]). Laypeople are unsure about the true meaning of the <emph>p</emph>-value reported in the study. The <emph>p</emph>-value represents the probability of observing the data given the null hypothesis is true rather than the probability of the null hypothesis being true. Furthermore, a statistically significant result might provide an illusion of practical importance because a significant <emph>p-</emph>value offers no indication of the effect size. Although null hypothesis testing still remains the primary mode of inquiry, people are beginning to emphasize confidence intervals more frequently in social science research.</p> <p>A confidence interval can be formed by extending a certain number of standard errors around a point estimate. The point estimate is the middle point of the confidence interval. Positive and negative allowances on either side of the point estimate provide the margins of error. The margin of error is half the width of a confidence interval, and it indicates the precision of the confidence interval.</p> <p>Sample size has a considerable bearing on the precision of a confidence interval because it affects the interval width. The width of a confidence interval is indicative of its precision. A wide confidence interval shows a long range of plausible values for the population parameter. Such a confidence interval gives very little insight into the population parameter and is deemed imprecise. The narrower the confidence interval is, the more precise the interval estimate becomes. [<reflink idref="bib5" id="ref8">5</reflink>] commented that the width of a confidence interval is analogous to the statistical power of a significance test, and that large sample sizes help improve the precision of confidence intervals, much as they increase the statistical power in significance tests. It is intuitively easy to explain Cohen's comments because a large sample size reduces the standard error and, in turn, the margin of error. The margin of error—the half width of a confidence interval—is the standard error multiplied by a critical value (e.g., a critical <emph>t</emph> value). The larger the sample size is, the more likely we will obtain a small margin of error.</p> <p>Although the literature has extensive coverage of sample size and statistical power ([<reflink idref="bib4" id="ref9">4</reflink>]; [<reflink idref="bib14" id="ref10">14</reflink>]), there is a lack of accessible references on sample size choice for confidence intervals. Few researchers know how to choose an appropriate sample size to construct a sufficiently narrow confidence interval. This article is intended to introduce the Statistical Analysis System (SAS) proc power procedure to determine sample size for a confidence interval in the dependent and independent <emph>t</emph> test.</p> <hd id="AN0094643633-2">SAMPLE SIZE FOR THE CONFIDENCE INTERVAL OF THE MEAN</hd> <p>There have been several seminal papers on sample sizes and confidence intervals ([<reflink idref="bib3" id="ref11">3</reflink>]; [<reflink idref="bib7" id="ref12">7</reflink>]; [<reflink idref="bib8" id="ref13">8</reflink>]; [<reflink idref="bib11" id="ref14">11</reflink>]). The approach is to find a sufficiently large sample size to achieve a high probability of obtaining a narrow confidence interval. [<reflink idref="bib3" id="ref15">3</reflink>] created a coherent framework for determining sample sizes for confidence intervals. He used the probability of obtaining a narrow width conditional on the confidence interval including the population parameter. The notation for this probability is , where the half width of the confidence interval is <emph>w</emph>; the upper bound of the half width is <emph>U</emph>. The inequality means that the half width is less than or equal to a predetermined upper bound (i.e., the desired precision). The event <emph>I</emph> means that the confidence interval includes the population parameter (e.g., the mean or mean difference). The probability of the confidence interval including the population parameter is the confidence level or . The significance level, , is traditionally set at 5%. The probability is a conditional probability, and it is related to the unconditional probability of obtaining a narrow width in a confidence interval . The relationship between and can be represented in the following equation,</p> <p>(<reflink idref="bib1" id="ref16">1</reflink>)</p> <p>Graph</p> <p>The opposite event () means that the confidence interval does not include the population parameter. The probability represents the likelihood that the confidence interval fails to capture the value of the population parameter. The conditional probability of achieving a certain width conditional on the confidence interval not including the population parameter is . The conditional probability is of more interest than the conditional probability . We desire to obtain a narrow confidence interval only when it includes the population parameter. The conditional probability is named the power of the confidence interval in [<reflink idref="bib3" id="ref17">3</reflink>], and it is the default output in the SAS proc power procedure ([<reflink idref="bib15" id="ref18">15</reflink>], p. 3171). Although there are some differences in value between the conditional probability and the unconditional probability , the difference is usually very small. The two probabilities are numerically about the same, but they represent different underlying philosophies. [<reflink idref="bib12" id="ref19">12</reflink>] argued that a narrow confidence interval is desired only if it includes the population parameter. Both [<reflink idref="bib3" id="ref20">3</reflink>] paper and the SAS procedure seem to reflect the same view. Therefore, we will try to compute the conditional probability in the examples, using SAS.</p> <p>The SAS proc power procedure calculates the conditional probability or power of the confidence interval by default. The researcher needs to specify the desired upper bound, , on the half width and estimate the population standard deviation, . The SAS procedure will return the power for the confidence interval, given the sample size, or it will calculate the required sample size given a specified power for the confidence interval. In the following, we will show how to determine sample sizes for confidence intervals in the dependent <emph>t</emph> test and the independent <emph>t</emph> test.</p> <hd id="AN0094643633-3">DEPENDENT T TEST</hd> <p>The dependent <emph>t</emph> test typically involves a sample of matched pairs or a group of subjects that have been tested twice. For instance, a group of participants are measured on a performance criterion before and after they receive a certain treatment. The difference of the observations before and after the treatment is the outcome of interest. If the treatment does not change the participants' performance, their difference scores will be zero on average. This is the null hypothesis about the mean of the difference scores. The alternative hypothesis, on the contrary, states that the mean of the difference scores is not zero. In other words, the participants' performance after the treatment is no longer the same as that before the treatment. The dependent <emph>t</emph> test can be used to rule out chance as a plausible explanation of the before-and-after-treatment performance differential. We can also use a confidence interval to measure the actual change in mean performance on the criterion. The confidence interval of the mean difference score is</p> <p>(<reflink idref="bib2" id="ref21">2</reflink>)</p> <p>Graph</p> <p>where is the sample mean of the difference scores, is the <emph>t</emph> critical value, is the sample standard deviation of the difference scores, and is the sample size. The half width of the confidence interval is .</p> <p>We can use the following SAS code ([<reflink idref="bib15" id="ref22">15</reflink>]) to find an appropriate sample size for the desired power for a confidence interval:</p> <p></p> <ulist> <item> proc power;</item> <p></p> <item> onesamplemeans ci=t</item> <p></p> <item> alpha = 0.05</item> <p></p> <item> halfwidth =.1 to 1.0 by.1</item> <p></p> <item> stddev = 1</item> <p></p> <item> probwidth = 0.80</item> <p></p> <item> ntotal =.;</item> <p></p> <item> plot x=effect min=.1 max=1.0;</item> <p></p> </ulist> <p>• run;</p> <p>The proc power procedure takes several arguments. Among others, "ci=t" suggests that it is for sample size and confidence interval in the one sample <emph>t</emph> test. It should be noted that the dependent <emph>t</emph> test follows the same computational procedure as the one-sample <emph>t</emph> test. That is why the dependent <emph>t</emph> test is listed under one-sample <emph>t</emph> test in the SAS proc power procedure. However, there is a difference between the two <emph>t</emph> tests. The former treats the difference scores between two paired samples of data as the outcome; the latter just uses the outcome in one sample of data. The significance level "alpha" is typically set to.05. The "halfwidth" () indicates how narrow the confidence interval should be. The argument "stddev" () specifies the standard deviation of the difference scores. The population standard deviation can be set as unity without losing generality because only the ratio between the half width and the standard deviation matters in SAS computation. In other words, SAS internally represents the half width as a multiple of the standard deviation. Had the "halfwidth" and "stddev" been ".2" and "2", they would be equivalent to "halfwidth=.1" and "stddev=1". A user can divide the half width by the specified standard deviation to obtain a multiple factor. The user can then set "halfwidth" to the multiple factor and "stddev" to 1. This frees the outcome measure from the underlying metric and puts different widths on a metric free scale. The user can supply different desired widths to the argument "halfwidth". In the example SAS code, the desired half width starts at.10 and increases by.1 until it reaches 1.0. SAS will subsequently produce the required sample sizes for the different levels of precision () in the confidence interval. The argument "probwidth" specifies the desired power of confidence interval, that is, the probability of having the half width shorter than the desired bound conditional on the confidence interval including the population parameter. The argument "ntotal" is left blank with a missing value ".". As it is left blank, SAS will compute the required total sample size to attain the specified power of confidence interval. If the power of confidence interval is left blank (i.e., "probwidth=.") and the total sample is supplied with a value, say, "ntotal=100", SAS will then return the power of confidence interval for the desired width and the sample size of 100.</p> <p>Table 1 lists the desired width, the required sample size for the specified power of the confidence interval, and the actual power of the confidence interval. The actual power of the confidence interval may not exactly match the specified power of the confidence interval because the sample size cannot be a fractional number. The sample size can only change in one unit increments. Therefore, the actual power of the confidence interval may be close to.80 but may not be exactly equal to.80.</p> <p>We can use a study of weight loss to show how to calculate the power of a confidence interval in the dependent <emph>t</emph> test. For example, a researcher may be interested in studying the effects of a three-month exercise and diet intervention on weight loss. The patients' body mass indices (BMI kg/m<sups>2</sups>) are measured before and after they undergo the regimen of physical exercise and daily diet. A dependent <emph>t</emph> test can be computed to see if the BMI change is statistically significantly different from zero. A <emph>t</emph> statistic and its <emph>p</emph>-value can be obtained from the dependent <emph>t</emph> test. As the <emph>t</emph> statistic and its <emph>p</emph>-value do not indicate whether the change of BMI is practically important, the researcher will estimate the mean change in BMI by constructing a confidence interval. Suppose that the confidence interval is expected to be accurate to 2 kg/m<sups>2</sups> (i.e.,), and that the standard deviation of BMI changes is estimated to be 4 kg/m<sups>2</sups> (). The desired precision of the confidence interval is . It is equivalent to setting and in the SAS proc power procedure, which produces the required sample sizes in Table 1. The researcher needs to recruit 22 participants to obtain a confidence interval accurate to 2 kg/m<sups>2</sups> and achieve a power of.80.</p> <p>The "plot" statement is optional in the SAS proc power procedure. It uses the desired half width, , as the horizontal axis, which starts at.10 and runs continuously through 1.0. The required sample sizes are plotted on the vertical axis against the desired half width, while the power of the confidence interval is held at.80 (see Figure 1). The "plot" statement offers an easy way to explore the required sample sizes for varying degrees of precision in the confidence interval.</p> <p>TABLE 1 Sample Size and Power of Confidence Interval in the Dependent t Test</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>w=U/σ</td><td>Power of Confidence Interval</td><td>Sample Size <italic>n</italic></td></tr></thead><tbody><tr><td char=".">0.1</td><td char=".">.808</td><td>410</td></tr><tr><td char=".">0.2</td><td char=".">.812</td><td>110</td></tr><tr><td char=".">0.3</td><td char=".">.825</td><td>53</td></tr><tr><td char=".">0.4</td><td char=".">.818</td><td>32</td></tr><tr><td char=".">0.5</td><td char=".">.812</td><td>22</td></tr><tr><td char=".">0.6</td><td char=".">.843</td><td>17</td></tr><tr><td char=".">0.7</td><td char=".">.805</td><td>13</td></tr><tr><td char=".">0.8</td><td char=".">.827</td><td>11</td></tr><tr><td char=".">0.9</td><td char=".">.880</td><td>10</td></tr><tr><td char=".">1.0</td><td char=".">.803</td><td>8</td></tr><tr><td><italic>Note:</italic> The standard deviation, <p><inline-graphic href="hmpe_a_864657_o_ilm0040.gif" /></p>, is set to one. The power of confidence interval is defined as <p><inline-graphic href="hmpe_a_864657_o_ilm0041.gif" /></p>. The symbol <p><inline-graphic href="hmpe_a_864657_o_ilm0042.gif" /></p> is the half width of a confidence interval. The symbol <p><inline-graphic href="hmpe_a_864657_o_ilm0043.gif" /></p> stands for the upper bound on the margin of error of the confidence interval. The event <p><inline-graphic href="hmpe_a_864657_o_ilm0044.gif" /></p> means that the confidence interval includes the population parameter.</td></tr></tbody></table> </ephtml> </p> <p>Graph: FIGURE 1  Achieving power of the confidence interval in the dependent t test.</p> <hd id="AN0094643633-4">INDEPENDENT T TEST</hd> <p>A sufficient sample size can also be determined to obtain a narrow confidence interval on the two-mean difference in an independent <emph>t</emph> test, which is often used to compare the outcome performance between a treatment group and a control group. If there is any performance differential between the treatment and control groups, one can make a causal inference about the treatment effect. A confidence interval of the mean difference can be computed to measure the magnitude of the treatment effect:</p> <p>(<reflink idref="bib3" id="ref23">3</reflink>)</p> <p>Graph</p> <p>where is the sample mean on the outcome for the treatment group, is the sample mean on the outcome for the control group, and <emph>s</emph> is the pooled sample standard deviation with degrees of freedom . For planning purposes, we assume a balanced design that has equal numbers of participants in the treatment and control conditions (i.e., ). The margin of error or the half width of the confidence interval is .</p> <p>The following SAS code ([<reflink idref="bib15" id="ref24">15</reflink>]) produces the required sample sizes to achieve.80 in the probability of having the half width narrower than the upper bound given that the confidence interval includes the true mean difference:</p> <p></p> <ulist> <item> proc power;</item> <p></p> <item> twosamplemeans ci=diff</item> <p></p> <item> halfwidth =.1 to 1.0 by.1</item> <p></p> <item> stddev =1</item> <p></p> <item> probwidth =.80</item> <p></p> <item> ntotal =.;</item> <p></p> <item> plot x=effect min=.1 max=1.0;</item> <p></p> </ulist> <p>• run;</p> <p>We can set the standard deviation as unity and express the upper bound as a multiple of the standard deviation , which ranges from.1 to 1.0 with an increment of.1 in the argument "halfwidth". The keyword "twosamplemeans ci=diff" instructs the SAS program to compute the required sample size for the confidence interval based on the independent <emph>t</emph> test. The rest of the code is similar to that in the dependent <emph>t</emph> test.</p> <p>We can use a study of fitness training to illustrate how to calculate the power of a confidence interval in a two sample independent <emph>t</emph> test. For example, a researcher may want to conduct a two-group comparison study to examine the effect of the new fitness training program on maximum oxygen consumption (VO<subs>2</subs> max in ml/kg/min) among young women aged 20 to 24. The researcher intends to recruit a random sample of participants and randomly assign them either to the new fitness training program or to the regular fitness training program. An independent <emph>t</emph> test can be used to see whether the new fitness training has significantly changed the maximum oxygen consumption beyond that of the regular fitness training. The <emph>t</emph> test will result in a <emph>p</emph>-value, by which the researcher can discern whether the performance differential between the two training programs results from haphazard chance. However, the <emph>p</emph>-value does not indicate the actual size of the performance differential or the clinical importance of the performance differential. As the researcher is very interested in the size of the differential performance, he or she decides to compute a confidence interval for the performance differential in VO<subs>2</subs> max. Suppose that the researcher plans to have the mean difference in VO<subs>2</subs> max accurate to 2 ml/kg/min (i.e., ) in the confidence interval. The upper bound is a subjective criterion, and the researcher may change it according to the desired precision. In estimating the standard deviation, the researcher can use the fact that six standard deviations cover 99.9% of the range of a normal outcome ([<reflink idref="bib1" id="ref25">1</reflink>]). He or she can find in the literature that the maximum oxygen consumption in VO<subs>2</subs> ranges from 27 to 51 ml/kg/min for women aged 20 to 24 ([<reflink idref="bib17" id="ref26">17</reflink>]). Thus, the estimated population standard deviation is . Alternatively, the standard deviation can be obtained directly from the relevant literature if it is available. The desired precision is . Table 2 shows that the required total sample size will move the power of the confidence interval above.80 for the desired precision. Alternatively, the researcher can set "halfwidth =" to 2 and "stddev =" to 4 in the SAS code:</p> <p>TABLE 2 Total Sample Size N and Power of Confidence Interval in the Independent t Test</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>w=U/σ</td><td>Power of Confidence Interval</td><td>Total Sample Size <italic>n</italic></td></tr></thead><tbody><tr><td char=".">0.1</td><td char=".">.805</td><td>1586</td></tr><tr><td char=".">0.2</td><td char=".">.808</td><td>410</td></tr><tr><td char=".">0.3</td><td char=".">.829</td><td>190</td></tr><tr><td char=".">0.4</td><td char=".">.810</td><td>110</td></tr><tr><td char=".">0.5</td><td char=".">.834</td><td>74</td></tr><tr><td char=".">0.6</td><td char=".">.850</td><td>54</td></tr><tr><td char=".">0.7</td><td char=".">.804</td><td>40</td></tr><tr><td char=".">0.8</td><td char=".">.811</td><td>32</td></tr><tr><td char=".">0.9</td><td char=".">.881</td><td>28</td></tr><tr><td char=".">1.0</td><td char=".">.801</td><td>22</td></tr><tr><td><italic>Note:</italic> Equal sample size is assumed for the treatment and control conditions. The total sample is <italic>N = 2n</italic>, and <italic>n</italic> is the group size<italic>.</italic> The standard deviation, <p><inline-graphic href="hmpe_a_864657_o_ilm0060.gif" /></p>, is set to 1. The power of confidence interval is defined as <p><inline-graphic href="hmpe_a_864657_o_ilm0061.gif" /></p>. The symbol <p><inline-graphic href="hmpe_a_864657_o_ilm0062.gif" /></p> is the half width of a confidence interval. The symbol <p><inline-graphic href="hmpe_a_864657_o_ilm0063.gif" /></p>stands for the upper bound on the margin of error of the confidence interval. The event <p><inline-graphic href="hmpe_a_864657_o_ilm0064.gif" /></p> means that the confidence interval includes the population parameter.</td></tr></tbody></table> </ephtml> </p> <p></p> <ulist> <item> proc power;</item> <p></p> <item> twosamplemeans ci=diff</item> <p></p> <item> halfwidth =2</item> <p></p> <item> stddev =4</item> <p></p> <item> probwidth =.80</item> <p></p> <item> ntotal =.;</item> <p></p> </ulist> <p>• run;</p> <p>The result will be the same because the ratio determines the desired precision. Figure 2 can also be used to explore the required sample sizes for different levels of precision in the confidence interval for the mean difference in maximum oxygen consumption.</p> <p>Graph: FIGURE 2  Achieving power of the confidence interval in the independent t test.</p> <hd id="AN0094643633-5">DISCUSSION</hd> <p>The use of confidence intervals has received great attention in social science research. The interval estimate provides a more realistic view of the population parameter than does a single point estimate or a <emph>p</emph>-value. The precision of the confidence interval depends on its width. A narrow confidence interval shows precisely where the population parameter may lie, whereas a wide confidence interval is not very informative about the actual value of the population parameter. The sample size has great influence on the margin of error for the confidence interval. The larger the sample size is, the smaller the standard error and the margin of error.</p> <p>In planning for a confidence interval with a sufficiently narrow width, the researcher needs to consider the desired precision and the probability of achieving that precision. It is crucial to choose an appropriate sample size to obtain a high chance of achieving the desired precision. An insufficient sample size limits the chances of achieving the desired precision, whereas an unnecessarily large sample size wastes precious resources. Thus, an adequate sample size prevents unnecessary waste and yields good results.</p> <p>The SAS proc power procedures can be utilized to determine an appropriate sample size for the confidence interval in the dependent and independent <emph>t</emph> test. Although researchers in physical education and exercise science are accustomed to sample size determination for statistical power, few have used the SAS proc power procedure to select sample size for confidence intervals. This introductory article provides a needed update on sample size choice for confidence intervals, as sample size determination for confidence intervals follows a different paradigm than that for significance tests.</p> <p>Sample size based on statistical power may not be sufficient to achieve the desired precision for a confidence interval. A smaller sample size is usually required to obtain the desired statistical power in a significance test, compared with that for similar power for a confidence interval. If the sample size for statistical power is used to compute a confidence interval, there will not be a high chance of achieving the desired precision in the confidence interval ([<reflink idref="bib13" id="ref27">13</reflink>]). It is easy to explain this without using complicated mathematics. It requires less effort to probe the existence of an effect in hypothesis testing than to measure such an effect with satisfactory precision in a confidence interval. Hypothesis testing is analogous to the first step in ascertaining an effect, and the confidence interval represents one step further in our inquiry. Hence, it is important not to use the sample size based on statistical power for confidence intervals.</p> <p>This article addresses sample size for two-sided confidence intervals in the <emph>t</emph> tests for continuous outcomes. The results can be readily extended to a one-sided confidence interval, which occurs less often than its two-sided counterpart. A one-sided confidence interval uses the point estimate plus or minus the margin of error as the closed-end limit. The margin of error will be used in lieu of the half width in this case. The one-sided confidence interval uses a slightly different critical value for the margin of error. The users just need to insert an additional statement "sides=1" into the SAS proc power procedure.</p> <p>The aforementioned confidence intervals assume a continuous outcome in the analysis. They do not apply to the analysis which uses a categorical outcome. For instance, a population proportion or percentage implies a dichotomous outcome. Although the population proportion is the mean of a dichotomous outcome, the statistical analysis of a dichotomous outcome uses quite different procedures than regular <emph>t</emph> tests. There are specialized formulas for calculating the required sample size for the confidence interval for estimating a population proportion or percentage ([<reflink idref="bib1" id="ref28">1</reflink>], p. 126).</p> <p>In conclusion, the required sample sizes for confidence intervals depend on the outcome type and the statistical analysis. There is no universal way to compute the required sample sizes for confidence intervals. The computational procedures must match the outcome type and the test statistics. The current theory on sample sizes for confidence intervals is still less developed than that for statistical power. As more people use confidence intervals, it is hoped that there will be further developments in sample size choice for confidence intervals in other statistical analyses such as ANOVA and the general linear model.</p> <ref id="AN0094643633-6"> <title> REFERENCES </title> <blist> <bibl id="bib1" idref="ref16" type="bt">1</bibl> <bibtext> Agresti, A. and Finlay, B.2009. Statistical methods for the social sciences , 4th ed, Upper Saddle River, NJ: Pearson Prentice Hall.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref3" type="bt">2</bibl> <bibtext> Batterham, A. and Hopkins, W.2006. Making meaningful inferences about magnitudes. International Journal of Sports Physiology and Performance, 1: 50–57.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref11" type="bt">3</bibl> <bibtext> Beal, S. L.1989. Sample size determination for confidence intervals on the population mean and on the difference between two population means. Biometrics, 45: 969–977.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref9" type="bt">4</bibl> <bibtext> Cohen, J.1988. Statistical power analysis for the behavioral sciences, Hillsdale, NJ: Lawrence Erlbaum. (2nd ed.)</bibtext> </blist> <blist> <bibl id="bib5" idref="ref4" type="bt">5</bibl> <bibtext> Cohen, J.1994. The Earth is round (p <.05). American Psychologist, 49: 997–1003.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref5" type="bt">6</bibl> <bibtext> Cumming, G. and Fidler, F.2009. Confidence intervals: Better answers to better questions. Zeitschrift fuer Psychologie, 217: 15–26.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref12" type="bt">7</bibl> <bibtext> Hahn, G. and Meeker, W.1991. Statistical intervals: A guide for practitioners, New York, NY: Wiley.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref13" type="bt">8</bibl> <bibtext> Hsu, J. C.1988. Sample size computation for designing multiple comparison experiments. Computational Statistics and Data Analysis, 7: 79–91.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref6" type="bt">9</bibl> <bibtext> Hunter, J. E.1997. Needed: A ban on the significance test. Psychological Science, 8: 3–7.</bibtext> </blist> <blist> <bibtext> International Committee of Medical Journal Editors. 1988. Uniform requirements for manuscripts submitted to biomedical journals. Annals of Internal Medicine, 108: 258–265.</bibtext> </blist> <blist> <bibtext> Jiroutek, M. R., Muller, K. E., Kupper, L. L. and Stewart, P. W.2003. A new method for choosing sample size for confidence interval-based inferences. Biometrics, 59: 580–590.</bibtext> </blist> <blist> <bibtext> Lehmann, E.1959. Testing statistical hypotheses, New York, NY: Wiley.</bibtext> </blist> <blist> <bibtext> Liu, X.2013. Comparing sample size requirements for significance tests and confidence intervals. Counseling Outcome Research and Evaluation, 4: 3–12.</bibtext> </blist> <blist> <bibtext> Park, I. and Schutz, R. W.1999. "Quick and Easy" formulae for approximating statistical power in repeated measures ANOVA. Measurement in Physical Education and Exercise Science, 3: 249–270.</bibtext> </blist> <blist> <bibtext> Institute, SAS. 2004. "9.1.3 help and documentation". In SAS, Cary, NC: Author.</bibtext> </blist> <blist> <bibtext> Schmidt, F.1996. Statistical significance testing and cumulative knowledge in psychology: Implications for the training of researchers. Psychological Methods, 1: 115–129.</bibtext> </blist> <blist> <bibtext> Shvartz, E. and Reibold, R.1990. Aerobic fitness norms for males and females aged 6 to 75 years: A review. Aviation, Space and Environmental Medicine, 61: 3–11.</bibtext> </blist> <blist> <bibtext> Wilkinson, L.1999. Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54: 494–604.</bibtext> </blist> </ref> <aug> <p>By XiaofengSteven Liu; Brandon Loudermilk and Thomas Simpson</p> <p>Reported by Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib18" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib16" firstref="ref7"></nolink> <nolink nlid="nl4" bibid="bib14" firstref="ref10"></nolink> <nolink nlid="nl5" bibid="bib11" firstref="ref14"></nolink> <nolink nlid="nl6" bibid="bib15" firstref="ref18"></nolink> <nolink nlid="nl7" bibid="bib12" firstref="ref19"></nolink> <nolink nlid="nl8" bibid="bib17" firstref="ref26"></nolink> <nolink nlid="nl9" bibid="bib13" firstref="ref27"></nolink>
Header DbId: eric
DbLabel: ERIC
An: EJ1029741
AccessLevel: 3
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Introduction to Sample Size Choice for Confidence Intervals Based on 't' Statistics
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Liu%2C+Xiaofeng+Steven%22">Liu, Xiaofeng Steven</searchLink><br /><searchLink fieldCode="AR" term="%22Loudermilk%2C+Brandon%22">Loudermilk, Brandon</searchLink><br /><searchLink fieldCode="AR" term="%22Simpson%2C+Thomas%22">Simpson, Thomas</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Measurement+in+Physical+Education+and+Exercise+Science%22"><i>Measurement in Physical Education and Exercise Science</i></searchLink>. 2014 18(2):91-100.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 10
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2014
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Descriptive
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Sample+Size%22">Sample Size</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Confidence+Testing%22">Confidence Testing</searchLink><br /><searchLink fieldCode="DE" term="%22Intervals%22">Intervals</searchLink><br /><searchLink fieldCode="DE" term="%22Measurement+Techniques%22">Measurement Techniques</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Exercise%22">Exercise</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/1091367X.2013.864657
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1091-367X
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Sample size can be chosen to achieve a specified width in a confidence interval. The probability of obtaining a narrow width given that the confidence interval includes the population parameter is defined as the power of the confidence interval, a concept unfamiliar to many practitioners. This article shows how to utilize the Statistical Analysis System (SAS) proc power procedure to determine an appropriate sample size and achieve sufficient power for a specified confidence interval. Two examples in sport and exercise science are used to illustrate sample size determination for confidence intervals in the dependent and independent "t" tests. The relevant SAS code and output are provided with detailed annotations. As the use of confidence intervals becomes a more integral part of studies in sport and exercise science, the reporting and analysis of their power should be considered much like that of hypothesis tests.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: Ref
  Label: Number of References
  Group: RefInfo
  Data: 18
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2014
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1029741
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1029741
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/1091367X.2013.864657
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 10
        StartPage: 91
    Subjects:
      – SubjectFull: Sample Size
        Type: general
      – SubjectFull: Statistical Analysis
        Type: general
      – SubjectFull: Confidence Testing
        Type: general
      – SubjectFull: Intervals
        Type: general
      – SubjectFull: Measurement Techniques
        Type: general
      – SubjectFull: Computation
        Type: general
      – SubjectFull: Exercise
        Type: general
    Titles:
      – TitleFull: Introduction to Sample Size Choice for Confidence Intervals Based on 't' Statistics
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Liu, Xiaofeng Steven
      – PersonEntity:
          Name:
            NameFull: Loudermilk, Brandon
      – PersonEntity:
          Name:
            NameFull: Simpson, Thomas
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2014
          Identifiers:
            – Type: issn-print
              Value: 1091-367X
          Numbering:
            – Type: volume
              Value: 18
            – Type: issue
              Value: 2
          Titles:
            – TitleFull: Measurement in Physical Education and Exercise Science
              Type: main
ResultId 1