STATISTICS AND RESEARCH DESIGN
Inference from a sample mean—Part 1 Nikolaos Pandis, Associate Editor of Statistics and Research Design Bern, Switzerland, and Corfu, Greece
I
n a previous article, I discussed how we can practically construct a confidence interval that is likely to include the true population mean from our single sample by taking advantage of the properties of the sampling distribution. In many cases, we can make a skewed distribution symmetrical by transforming the data; this allows us to use the properties of the sampling distribution. The standard error (SE) of the sample mean is the estimated standard deviation of the sampling distribution; because we carry out the study only once, we have only 1 sample. Even though we do not observe the sampling distribution, since this would require multiple samples, we can calculate the standard deviation of the sampling distributiongiven the standard deviation s and size of our 1 sample pffiffi . n
Again, using information from this 1 sample, along with knowledge about the sampling distribution of a mean, we can construct a confidence interval. As long as the size of our sample is reasonably large (.30), we know that the sampling distribution of a mean is normal. This allows us to infer a range of values within which the true population mean is likely to lie. This range of values is called a “confidence interval” because we can be reasonably confident that it contains the true mean. I described earlier that the confidence interval of the mean follows the general formula: mean 6 1:96 3 SE 5 14:5 6 1:96 3 SE If we took thousands of samples and for each sample calculated the mean and associated 95% confidence interval, we would expect 95% of these confidence intervals to include the population mean. When we are making statistical comparisons, we state the null (Ho) and the alternative (Ha) hypotheses. The null hypothesis usually states that there is no difference when comparing groups, and we look for evidence to disprove the null hypothesis and accept the Visiting assistant professor, Department of Orthodontics and Dentofacial Orthopedics, Faculty of Medicine, University of Bern, Bern, Switzerland; private practice, Corfu, Greece. Am J Orthod Dentofacial Orthop 2015;147:791-3 0889-5406/$36.00 Copyright Ó 2015 by the American Association of Orthodontists. http://dx.doi.org/10.1016/j.ajodo.2015.03.015
alternative. In our previous example of adolescent patients seeking orthodontic therapy, our sample's mean age was 14.5 years, and we hypothesized that our true population mean (m) was 15.3 years. We are interested in assessing whether our assumption about the population mean holds. In other words, we would like to test whether our sample mean differs from the hypothesized true population mean. To accomplish this, we compare 2 hypotheses: (1) the null hypothesis that the population mean is equal to m, H0:m0 5 m; and (2) the alternative hypothesis that the population mean is any value except for m, H1: m1 sm. The statistical process to assess the null hypothesis is called the statistical test. Since the alternative hypothesis is defined as 2-sided, the statistical test is a 2-sided test. The 2-sided test is applied more frequently than the “strict” 1-sided test (eg, H1:m1.m). Hypothesis testing
This step allows us to calculate a statistical formula known as the test statistic: a measure of how extreme our data are. The general form of the test statistic compares the observed value estimated from the sample, x; and the population mean under the null hypothesis, m0 5 m. It also takes into account the population variability by using the standard error. We saw that sampling distributions are approximately normal when the sample size is large. The letter “z” is often used in association with normal distribution, so the value of the test statistic is called a “z value.” It is equal to x m0 SE between the sample mean and the population mean (under the null hypothesis), and it is a measure of how far the observed mean is from the population mean under the null hypothesis, expressed in units of standard errors. Hypothesis tests that make use of a z value are generally called “z tests.” In our example, we are making inferences based on 1 sample (comparing the mean [14.5 years] from our sample of adolescent orthodontic patients with the hypothesized true population mean [15.3 years] of adolescent orthodontic patients), so we call this a 1-sample z test. If the z 5
791
Statistics and research design
792
Fig. Comparison of z (left) and t (right) distributions.
null hypothesis is true, all the many values of z that could be obtained from different samples taken from the same population would be distributed as a standard normal distribution. In our example, the sample mean is normally distributed with a mean of m 5 14.5 and a standard error of 5:79 ffiffiffiffi 5 0:67. SE 5 pSDffiffin 5 p 74 The assumptions to implement a z test are that (1) the population is normally distributed, (2) the population's standard deviation is known, and (3) the sample size is large enough (.30).1 When assumptions 2 and 3 do not hold, then another test is used, called the “t test.” The implementation of this test dictates the estimation of the sample's standard deviation under the formula qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 S5 i 5 1 ðxi xÞ =ðn 1Þ, where x is the sample mean and n is the sample size. Then, a t value can be calculated as follows: t5
x m0 pffiffiffi S= n
If the null hypothesis is true, all the many t values that could be obtained from different samples from the same population would be distributed as a t distribution (also known as “Student t distribution”). The t distribution has a bell-like shape centered at zero (like the standard normal distribution), but the bell is smaller and broader for small samples. The shape of the t distribution depends on its degrees of freedom (df), a measure of how much of the sample size is used to determine the t distribution. In the 1-sample case, the degrees of
June 2015 Vol 147 Issue 6
freedom of a t distribution are equal to the sample size minus 1. The degrees of freedom depend on the sample size. As the sample size increases, the degrees of freedom increase; hence, according to the central limit theorem, the t distribution looks more and more like the standard normal distribution. So, when the sample size is small, say less than 30, the formula for a 95% confidence interval for a mean becomes the estimated mean 6 t1 20:95;n 1 3 SE, where t1 20:95;n 1 is a multiplier known as the upper 0.025 percentile point on the t distribution with n 1 df. In other words, when the sample size is large and we use the z distribution, the multiplier is always 1.96. However, when we use smaller sample sizes and apply the t distribution, the multiplier varies depending on the sample size and tends to reach 1.96 from above as the sample size increases. The result of the larger multiplier is that the calculated 95% confidence interval from the t distribution is larger than the confidence interval resulting from the z distribution. The wider confidence interval accounts for the extra uncertainty from the small sample. For hypothesis testing, the formula is the same, but the P value must be read from tables of the t distribution. The t distribution has critical values that are larger than the z distribution to account for the extra uncertainty because of the small sample. The t distribution approximates well the z distribution when the sample size is larger than 30. The Figure shows that the multiplier from the t distribution with 20 df is 2.09 compared with 1.96 of the z distribution.
American Journal of Orthodontics and Dentofacial Orthopedics
Statistics and research design
In the next article, Part 2, I will discuss how to obtain the P value and complete the statistical testing example.
793
REFERENCE 1. Kirkwood BR, Sterne JAC. Essential medical statistics. 2nd ed. Oxford, United Kingdom: Blackwell; 2003. p. 58-70.
Nostalgia Advertisement from a 1997 issue of the Journal
American Journal of Orthodontics and Dentofacial Orthopedics
June 2015 Vol 147 Issue 6
Inference from a sample mean—Part 1 Nikolaos Pandis, Associate Editor of Statistics and Research Design Bern, Switzerland, and Corfu, Greece
I
n a previous article, I discussed how we can practically construct a confidence interval that is likely to include the true population mean from our single sample by taking advantage of the properties of the sampling distribution. In many cases, we can make a skewed distribution symmetrical by transforming the data; this allows us to use the properties of the sampling distribution. The standard error (SE) of the sample mean is the estimated standard deviation of the sampling distribution; because we carry out the study only once, we have only 1 sample. Even though we do not observe the sampling distribution, since this would require multiple samples, we can calculate the standard deviation of the sampling distributiongiven the standard deviation s and size of our 1 sample pffiffi . n
Again, using information from this 1 sample, along with knowledge about the sampling distribution of a mean, we can construct a confidence interval. As long as the size of our sample is reasonably large (.30), we know that the sampling distribution of a mean is normal. This allows us to infer a range of values within which the true population mean is likely to lie. This range of values is called a “confidence interval” because we can be reasonably confident that it contains the true mean. I described earlier that the confidence interval of the mean follows the general formula: mean 6 1:96 3 SE 5 14:5 6 1:96 3 SE If we took thousands of samples and for each sample calculated the mean and associated 95% confidence interval, we would expect 95% of these confidence intervals to include the population mean. When we are making statistical comparisons, we state the null (Ho) and the alternative (Ha) hypotheses. The null hypothesis usually states that there is no difference when comparing groups, and we look for evidence to disprove the null hypothesis and accept the Visiting assistant professor, Department of Orthodontics and Dentofacial Orthopedics, Faculty of Medicine, University of Bern, Bern, Switzerland; private practice, Corfu, Greece. Am J Orthod Dentofacial Orthop 2015;147:791-3 0889-5406/$36.00 Copyright Ó 2015 by the American Association of Orthodontists. http://dx.doi.org/10.1016/j.ajodo.2015.03.015
alternative. In our previous example of adolescent patients seeking orthodontic therapy, our sample's mean age was 14.5 years, and we hypothesized that our true population mean (m) was 15.3 years. We are interested in assessing whether our assumption about the population mean holds. In other words, we would like to test whether our sample mean differs from the hypothesized true population mean. To accomplish this, we compare 2 hypotheses: (1) the null hypothesis that the population mean is equal to m, H0:m0 5 m; and (2) the alternative hypothesis that the population mean is any value except for m, H1: m1 sm. The statistical process to assess the null hypothesis is called the statistical test. Since the alternative hypothesis is defined as 2-sided, the statistical test is a 2-sided test. The 2-sided test is applied more frequently than the “strict” 1-sided test (eg, H1:m1.m). Hypothesis testing
This step allows us to calculate a statistical formula known as the test statistic: a measure of how extreme our data are. The general form of the test statistic compares the observed value estimated from the sample, x; and the population mean under the null hypothesis, m0 5 m. It also takes into account the population variability by using the standard error. We saw that sampling distributions are approximately normal when the sample size is large. The letter “z” is often used in association with normal distribution, so the value of the test statistic is called a “z value.” It is equal to x m0 SE between the sample mean and the population mean (under the null hypothesis), and it is a measure of how far the observed mean is from the population mean under the null hypothesis, expressed in units of standard errors. Hypothesis tests that make use of a z value are generally called “z tests.” In our example, we are making inferences based on 1 sample (comparing the mean [14.5 years] from our sample of adolescent orthodontic patients with the hypothesized true population mean [15.3 years] of adolescent orthodontic patients), so we call this a 1-sample z test. If the z 5
791
Statistics and research design
792
Fig. Comparison of z (left) and t (right) distributions.
null hypothesis is true, all the many values of z that could be obtained from different samples taken from the same population would be distributed as a standard normal distribution. In our example, the sample mean is normally distributed with a mean of m 5 14.5 and a standard error of 5:79 ffiffiffiffi 5 0:67. SE 5 pSDffiffin 5 p 74 The assumptions to implement a z test are that (1) the population is normally distributed, (2) the population's standard deviation is known, and (3) the sample size is large enough (.30).1 When assumptions 2 and 3 do not hold, then another test is used, called the “t test.” The implementation of this test dictates the estimation of the sample's standard deviation under the formula qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 S5 i 5 1 ðxi xÞ =ðn 1Þ, where x is the sample mean and n is the sample size. Then, a t value can be calculated as follows: t5
x m0 pffiffiffi S= n
If the null hypothesis is true, all the many t values that could be obtained from different samples from the same population would be distributed as a t distribution (also known as “Student t distribution”). The t distribution has a bell-like shape centered at zero (like the standard normal distribution), but the bell is smaller and broader for small samples. The shape of the t distribution depends on its degrees of freedom (df), a measure of how much of the sample size is used to determine the t distribution. In the 1-sample case, the degrees of
June 2015 Vol 147 Issue 6
freedom of a t distribution are equal to the sample size minus 1. The degrees of freedom depend on the sample size. As the sample size increases, the degrees of freedom increase; hence, according to the central limit theorem, the t distribution looks more and more like the standard normal distribution. So, when the sample size is small, say less than 30, the formula for a 95% confidence interval for a mean becomes the estimated mean 6 t1 20:95;n 1 3 SE, where t1 20:95;n 1 is a multiplier known as the upper 0.025 percentile point on the t distribution with n 1 df. In other words, when the sample size is large and we use the z distribution, the multiplier is always 1.96. However, when we use smaller sample sizes and apply the t distribution, the multiplier varies depending on the sample size and tends to reach 1.96 from above as the sample size increases. The result of the larger multiplier is that the calculated 95% confidence interval from the t distribution is larger than the confidence interval resulting from the z distribution. The wider confidence interval accounts for the extra uncertainty from the small sample. For hypothesis testing, the formula is the same, but the P value must be read from tables of the t distribution. The t distribution has critical values that are larger than the z distribution to account for the extra uncertainty because of the small sample. The t distribution approximates well the z distribution when the sample size is larger than 30. The Figure shows that the multiplier from the t distribution with 20 df is 2.09 compared with 1.96 of the z distribution.
American Journal of Orthodontics and Dentofacial Orthopedics
Statistics and research design
In the next article, Part 2, I will discuss how to obtain the P value and complete the statistical testing example.
793
REFERENCE 1. Kirkwood BR, Sterne JAC. Essential medical statistics. 2nd ed. Oxford, United Kingdom: Blackwell; 2003. p. 58-70.
Nostalgia Advertisement from a 1997 issue of the Journal
American Journal of Orthodontics and Dentofacial Orthopedics
June 2015 Vol 147 Issue 6
