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The degrees of freedom characterize the shape of the *t*-distribution. The larger the degrees of freedom, the more closely the distribution approximates the normal model.
Probabilities for the *t*-distribution can be calculated either by using distribution tables or using statistical software. The use of software has... | {
"Header 1": "**5.1 Single-sample inference with the** *t***-distribution**",
"Header 3": "**DEGREES OF FREEDOM (DF)**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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A *t*-distribution with 20 degrees of freedom is shown in the left panel of Figure [5.5.](#page-239-1) Estimate the proportion of the distribution falling above 1.65 and below -1.65.
Identify the row in the *t*-table using the degrees of freedom: *df* −20. Then, look for 1.65; the value is not listed, and falls betwe... | {
"Header 1": "**5.1 Single-sample inference with the** *t***-distribution**",
"Header 3": "**EXAMPLE 5.2**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Chapter [4](#page-197-0) provided formulas for tests and confidence intervals for population means in random samples large enough for the *t*-statistic to have a nearly normal distribution. In samples smaller than 30 from approximately symmetric distributions without large outliers, the *t*-statistic has a *t*-distribu... | {
"Header 1": "**5.1 Single-sample inference with the** *t***-distribution**",
"Header 3": "**5.1.2 Using the** *t***-distribution for tests and confidence intervals for a population mean**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Dolphins are at the top of the oceanic food chain; as a consequence, dangerous substances such as mercury tend to be present in their organs and muscles at high concentrations. In areas where dolphins are regularly consumed, it is important to monitor dolphin mercury levels. This example uses data from a random sample ... | {
"Header 1": "**5.1 Single-sample inference with the** *t***-distribution**",
"Header 3": "**EXAMPLE 5.3**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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According to the EPA, regulatory action should be taken if fish species are found to have a mercury level of 0.5 ppm or higher. Conduct a formal significance test to evaluate whether the average mercury content of croaker white fish (Pacific) is different from 0.50 ppm. Use $\alpha = 0.05$ .
The FDA regulatory guide... | {
"Header 1": "**5.1 Single-sample inference with the** *t***-distribution**",
"Header 3": "**EXAMPLE 5.5**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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In the 2000 Olympics, was the use of a new wetsuit design responsible for an observed increase in swim velocities? In a study designed to investigate this question, twelve competitive swimmers swam 1500 meters at maximal speed, once wearing a wetsuit and once wearing a regular swimsuit.[5](#page-243-1) The order of wet... | {
"Header 1": "**5.2 Two-sample test for paired data**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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When testing a hypothesis about paired data, compare the groups by testing whether the population mean of the differences between the groups equals 0.
- For a two-sided test, *H*<sup>0</sup> : *δ* = 0; *H<sup>A</sup>* : *δ* , 0.
- For a one-sided test, either *H*<sup>0</sup> : *δ* = 0; *H<sup>A</sup>* : *δ >* 0 or *H... | {
"Header 1": "**5.2 Two-sample test for paired data**",
"Header 3": "**STATING HYPOTHESES FOR PAIRED DATA**",
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Some important assumptions are being made. First, it is assumed that the data are a random sample from the population. While the observations are likely independent, it is more difficult to justify that this sample of 12 swimmers is randomly drawn from the entire population of competitive swimmers. Nevertheless, it is ... | {
"Header 1": "**5.2 Two-sample test for paired data**",
"Header 3": "5.2. TWO-SAMPLE TEST FOR PAIRED DATA 245",
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Using the data in Figure [5.7,](#page-243-2) conduct a two-sided hypothesis test at *α* = 0*.*05 to assess whether there is evidence to suggest that wetsuits have an effect on swim velocities during a 1500m swim.
The hypotheses are *H*<sup>0</sup> : *δ* = 0 and *H<sup>A</sup>* : *δ* , 0. Let *α* = 0*.*05.
Calculate... | {
"Header 1": "**5.2 Two-sample test for paired data**",
"Header 3": "**EXAMPLE 5.6**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Does treatment using embryonic stem cells (ESCs) help improve heart function following a heart attack? New and potentially risky treatments are sometimes tested in animals before studies in humans are conducted. In a 2005 paper in *Lancet*, Menard, et al. describe an experiment in which 18 sheep with induced heart atta... | {
"Header 1": "**5.3 Two-sample test for independent data**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Figure [5.9](#page-246-2) contains summary statistics for the 18 sheep.[10](#page-246-3) Percent change in heart pumping capacity was measured for each sheep. A positive value corresponds to increased pumping capacity, which generally suggests a stronger recovery from the heart attack. Is there evidence for a potential... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "**5.3.1 Confidence interval for a difference of means**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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The *t*-distribution can be used for inference when working with the standardized difference of two means if (1) each sample meets the conditions for using the *t*-distribution and (2) the samples are independent.
A confidence interval for a difference of two means has the same basic structure as previously discussed... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "**USING THE** *t***-DISTRIBUTION FOR A DIFFERENCE IN MEANS**",
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(E)
Calculate and interpret a 95% confidence interval for the effect of ESCs on the change in heart pumping capacity of sheep following a heart attack.
The point estimate for the difference is $\overline{x}_1 - \overline{x}_2 = \overline{x}_{esc} - \overline{x}_{control} = 7.83$ .
The standard error is:
$$\sqr... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "**EXAMPLE 5.9**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Is there evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who do not smoke? The dataset births contains data from a random sample of 150 cases of mothers and their newborns in North Carolina over a year; there are 50 cases in the smoking group and 100 cases ... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "5.3.2 Hypothesis tests for a difference in means",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Evaluate whether it is appropriate to apply the *t*-distribution to the difference in sample means between the two groups.
Since the data come from a simple random sample and consist of less than 10% of all such cases, the observations are independent. While each distribution is strongly skewed, the large sample size... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "**EXAMPLE 5.10**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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When testing a hypothesis about two independent groups, directly compare the two population means and state hypotheses in terms of *µ*<sup>1</sup> and *µ*2.
- For a two-sided test, *H*<sup>0</sup> : *µ*<sup>1</sup> = *µ*2; *H<sup>A</sup>* : *µ*<sup>1</sup> , *µ*2.
- For a one-sided test, either *H*<sup>0</sup> : *µ*<... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "**STATING HYPOTHESES FOR TWO-GROUP DATA**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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In this setting, the formula for a *t*-statistic is:
$$t = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{SE_{\overline{x}_1 - \overline{x}_2}} = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}.$$
Under the null hypothesis of no difference between... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "5.3. TWO-SAMPLE TEST FOR INDEPENDENT DATA 251",
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Using Figure [5.13,](#page-250-0) conduct a hypothesis test to evaluate whether there is evidence that newborns from mothers who smoke have a different average birth weight than newborns from mothers who do not smoke.
The hypotheses are *H*<sup>0</sup> : *µ*<sup>1</sup> = *µ*<sup>2</sup> and *H<sup>A</sup>* : *µ*<sup... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "**EXAMPLE 5.11**",
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In the two-sample setting, students often find it difficult to determine whether a paired test or an independent group test should be used. The paired test applies only in situations where there is a natural pairing of observations between groups, such as in the swim data. Pairing can be obvious, such as the two measur... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "5.3.3 The paired test vs. independent group test",
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When ignoring age, expenditures within the ethnicity groups Hispanic and White non-Hispanic show substantial right-skewing (Figure [1.45\)](#page-51-5). A transformation is advisable before conducting a *t*-test. As shown in Figure [5.14,](#page-252-0) a natural log transformation effectively eliminates skewing.
![](... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "Comparing expenditures overall",
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One way to account for the effect of age is to compare mean expenditures within age cohorts. When comparing individuals of similar ages but different ethnic groups, are the differences in mean expenditures larger than would be expected by chance alone?
Figure [1.52](#page-55-0) shows that the age cohort 13-17 is the ... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "Comparing expenditures within age cohorts",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Figure [5.17](#page-254-0) contains the summary statistics for computing the test statistic to compare expenditures in the two groups within this age cohort. The test statistic has value *t* = 0*.*318, with degrees of freedom 66. The two-sided *p*-value is 0.75. There is not evidence of a difference between mean expend... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "5.3. TWO-SAMPLE TEST FOR INDEPENDENT DATA 255",
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Occasionally, two populations will have standard deviations that are so similar that they can be treated as identical. For example, historical data or a well-understood biological mechanism may justify this strong assumption. In such cases, it can be more precise to use a pooled standard deviation to make inferences ab... | {
"Header 1": "**5.3 Two-sample test for independent data**",
"Header 3": "5.3.5 Pooled standard deviation estimate",
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Designing a study often involves many complex issues; perhaps the most important statistical issue in study design is the choice of an appropriate sample size. The power of a statistical test is the probability that the test will reject the null hypothesis when the alternative hypothesis is true; sample sizes are chose... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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A company would like to run a clinical trial with participants whose systolic blood pressures are between 140 and 180 mmHg. Suppose previously published studies suggest that the standard deviation of patient blood pressures will be about 12 mmHg, with an approximately symmetric distribution.[18](#page-256-2) What would... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "**EXAMPLE 5.13**",
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For what values of *x*trmt − *x*ctrl would the null hypothesis be rejected, using *α* = 0*.*05?
If the observed difference is in the far left or far right tail of the null distribution, there is sufficient evidence to reject the null hypothesis. For *α* = 0*.*05, *H*<sup>0</sup> is rejected if the difference is in th... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "**EXAMPLE 5.14**",
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Suppose the study proceeded with 100 patients per treatment group and the new drug does reduce average blood pressure by an additional 3 mmHg relative to the standard medication. What is the probability of detecting this effect?
Determine the sampling distribution for $\overline{x}_{trmt} - \overline{x}_{ctrl}$ whe... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "EXAMPLE 5.15",
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The last example demonstrated that with a sample size of 100 in each group, there is a probability of about 0.42 of detecting an effect size of 3 mmHg. If the study were conducted with this sample size, even if the new medication reduced blood pressure by 3 mmHg compared to the control group, there is a less than 50% c... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "5.4.3 Determining a proper sample size",
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Calculate the power to detect a change of -3 mmHg using a sample size of 500 per group. Recall that the standard deviation of patient blood pressures was expected to be about 12 mmHg.<sup>20</sup>
- (G)
- (a) Determine the standard error.
- (b) Identify the null distribution and rejection regions, as well as the alte... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "**GUIDED PRACTICE 5.16**",
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Identify the sample size that would lead to a power of 80%.
The *Z*-score that defines a lower tail area of 0.80 is about *Z* = 0*.*84. In other words, 0.84 standard errors from -3, the mean of the alternative distribution.
For *α* = 0*.*05, the rejection region always extends 1.96 st... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "**EXAMPLE 5.17**",
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Suppose the targeted power is 90% and *α* = 0*.*01. How many standard errors should separate the centers of the null and alternative distributions, where the alternative distribution is centered at the minimum effect size of interest? Assume the test is two-sided.[21](#page-260-0)
<sup>21</sup>Find the *Z*-score such... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "**GUIDED PRACTICE 5.18**",
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The previous sections have illustrated how power and sample size can be calculated from first principles, using the fundamental ideas behind distributions and testing. In practice, power and sample size calculations are so important that statistical software should be the method of choice; there are many commercially a... | {
"Header 1": "**5.4 Power calculations for a difference of means**",
"Header 3": "**5.4.4 Formulas for power and sample size**",
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In some settings, it is useful to compare means across several groups. It might be tempting to do pairwise comparisons between groups; for example, if there are three groups (*A,B,C*), why not conduct three separate *t*-tests (*A* vs. *B*, *A* vs. *C*, *B* vs. *C*)? Conducting multiple tests on the same data increases ... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Examine Figure [5.22.](#page-263-1) Compare groups I, II, and III. Is it possible to visually determine if the differences in the group centers is due to chance or not? Now compare groups IV, V, and VI. Do the differences in these group centers appear to be due to chance?
It is difficult to discern a difference in th... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "**EXAMPLE 5.19**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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The null hypothesis under consideration is the following: *µ*CC = *µ*CT = *µ*TT. Write the null and corresponding alternative hypotheses in plain language.[22](#page-264-0)
Figure [5.23](#page-264-1) provides summary statistics for each group. A side-by-side boxplot for the change in non-dominant arm strength is show... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "**GUIDED PRACTICE 5.20**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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(E)
The largest difference between the sample means is between the CC and TT groups. Consider again the original hypotheses:
- $H_0: \ \mu_{CC} = \mu_{CT} = \mu_{TT}$
- $H_A$ : The average percent change in non-dominant arm strength ( $\mu_i$ ) varies across some (or all) groups.
Why might it be inappropriate to ... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "**EXAMPLE 5.21**",
"token_count": 465,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Under the null hypothesis, any observed variation in group means is due to chance and there is no real difference between the groups. In other words, the null hypothesis assumes that the groupings are non-informative, such that all observations can be thought of as belonging to a single group. If this scenario is true,... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "5.5. COMPARING MEANS WITH ANOVA 267",
"token_count": 340,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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Calculate the *F*-statistic for the famuss data summarized in Figure [5.23.](#page-264-1) The overall mean *x* across all observations is 53.29.
First, calculate the *MSG* and *MSE*.
$$MSG = \frac{1}{k-1} \sum_{i=1}^{k} n_i (\bar{x}_i - \bar{x})^2$$
= $\frac{1}{3-1} [(173)(48.89 - 53.29)^2 + (261)(53.25 - 53.29)... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "**EXAMPLE 5.23**",
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The calculations required to perform an ANOVA by hand are tedious and prone to human error. Instead, it is common to use statistical software to calculate the *F*-statistic and associated p-value. The results of an ANOVA can be summarized in a table similar to that of a regression summary, which will be discussed in Ch... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "5.5.2 Reading an ANOVA table from software",
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Rejecting the null hypothesis in an ANOVA analysis only allows for a conclusion that there is evidence for a difference in group means. In order to identify the groups with different means, it is necessary to perform further testing. For example, in the famuss analysis, there are three comparisons to make: CC to CT, CC... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "5.5.3 Multiple comparisons and controlling Type I Error rate",
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The ANOVA conducted on the famuss dataset showed strong evidence of differences in the mean strength change in the non-dominant arm between the three genotypes. Complete the three possible pairwise comparisons using the Bonferroni correction and report any differences.
Use a modified significance level of $\alpha^* ... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "EXAMPLE 5.25",
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Statistical software can be used to calculate the *p*-values associated with each possible pairwise comparison of the groups in ANOVA. The results of the pairwise tests are summarized in a table that shows the *p*-value for each two-group test.
Figure [5.28](#page-270-0) shows the *p*-values from the three possible t... | {
"Header 1": "**5.5 Comparing means with ANOVA**",
"Header 3": "**5.5.4 Reading the results of pairwise** *t***-tests from software**",
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The material in this chapter is particularly important. For many applications, *t*-tests and Analysis of Variance (ANOVA) are an essential part of the core of statistics in medicine and the life sciences. The comparison of two or more groups is often the primary aim of experiments both in the laboratory and in studies ... | {
"Header 1": "**5.6 Notes**",
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Finally, the formula used to approximate degrees of freedom *ν* for the independent two-group *t*-test that does not assume equal variance is
$$\nu = \frac{\left[(s_1^2/n_1) + (s_2^2/n_2)\right]^2}{\left[(s_1^2/n_1)^2/(n_1-1) + (s_2^2/n_2)^2/(n_2-1)\right]},$$
where *n*1*, s*<sup>1</sup> are the sample size and sta... | {
"Header 1": "**5.6 Notes**",
"Header 3": "5.6. NOTES 273",
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**[5.14](#page-448-3) Air quality.** Air quality measurements were collected in a random sample of 25 country capitals in 2013, and then again in the same cities in 2014. We would like to use these data to compare average air quality between the two years. Should we use a paired or non-paired test? Explain your reasoni... | {
"Header 1": "**5.7 Exercises**",
"Header 3": "**5.7.2 Two-sample test for paired data**",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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**[5.21](#page-448-10) Gifted children.** Researchers collected a simple random sample of 36 children who had been identified as gifted in a large city. The following histograms show the distributions of the IQ scores of mothers and fathers of these children. Also provided are some sample statistics.[27](#page-278-0) ... | {
"Header 1": "**5.7 Exercises**",
"Header 3": "5.7. EXERCISES 279",
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**[5.26](#page-449-3) Egg volume.** In a study examining 131 collared flycatcher eggs, researchers measured various characteristics in order to study their relationship to egg size (assayed as egg volume, in *mm*<sup>3</sup> ). These characteristics included nestling sex and survival. A single pair of collared flycatch... | {
"Header 1": "**5.7 Exercises**",
"Header 3": "5.7. EXERCISES 281",
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"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
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**[5.31](#page-450-0) Chicken diet and weight, Part II.** Casein is a common weight gain supplement for humans. Does it have an effect on chickens? Using data provided in Exercise [5.29,](#page-281-2) test the hypothesis that the average weight of chickens that were fed casein is different than the average weight of ch... | {
"Header 1": "**5.7 Exercises**",
"Header 3": "5.7. EXERCISES 283",
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**[5.36](#page-450-5) Email outreach efforts.** A medical research group is recruiting people to complete short surveys about their medical history. For example, one survey asks for information on a person's family history in regards to cancer. Another survey asks about what topics were discussed during the person's la... | {
"Header 1": "**5.7 Exercises**",
"Header 3": "**5.7.4 Power calculations for a difference of means**",
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**[5.47](#page-451-2) Prison isolation experiment, Part II.** Exercise [5.35](#page-283-1) introduced an experiment that was conducted with the goal of identifying a treatment that reduces subjects' psychopathic deviant T scores, where this score measures a person's need for control or his rebellion against control. In... | {
"Header 1": "**5.7 Exercises**",
"Header 3": "5.7. EXERCISES 289",
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- **[6.1](#page-292-0) [Examining scatterplots](#page-292-0)**
- **[6.2](#page-294-0) [Estimating a regression line using least squares](#page-294-0)**
- **[6.3](#page-297-0) [Interpreting a linear model](#page-297-0)**
- **[6.4](#page-307-0) [Statistical inference with regression](#page-307-0)**
- **[6.5](#page-311-0)... | {
"Header 1": "**Simple linear regression**",
"token_count": 881,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Various demographic and cardiovascular risk factors were collected as a part of the Prevention of REnal and Vascular END-stage Disease (PREVEND) study, which took place in the Netherlands. The initial study population began as 8,592 participants aged 28-75 years who took a first survey in 1997-1998.<sup>3</sup> Partici... | {
"Header 1": "6.1 Examining scatterplots",
"token_count": 903,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Figure [6.3](#page-294-1) shows the scatterplot of age versus RFFT score, with the least squares regression line added to the plot; this line can also be referred to as a linear model for the data. An RFFT score can be predicted for a given age from the equation of the regression line:
$$\widehat{\text{RFFT}} = 137.5... | {
"Header 1": "**6.2 Estimating a regression line using least squares**",
"token_count": 422,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The residual of the *i th* observation (*x<sup>i</sup> ,yi* ) is the difference of the observed response (*y<sup>i</sup>* ) and the response predicted based on the model fit (b*y<sup>i</sup>* ):
$$e_i = y_i - \widehat{y_i}$$
The value <sup>b</sup>*y<sup>i</sup>* is calculated by plugging *x<sup>i</sup>* into the mo... | {
"Header 1": "**6.2 Estimating a regression line using least squares**",
"Header 3": "**RESIDUAL: DIFFERENCE BETWEEN OBSERVED AND EXPECTED**",
"token_count": 911,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
From the summary statistics displayed in Figure 6.4 for prevend.samp, calculate the equation of the least-squares regression line for the PREVEND data.
$$b_1 = \frac{s_y}{s_x}r = \frac{27.40}{11.60}(-0.534) = -1.26$$
$$b_0 = \overline{y} - b_1 \overline{x} = 68.40 - (-1.26)(54.82) = 137.55.$$
The results agree wi... | {
"Header 1": "**6.2 Estimating a regression line using least squares**",
"Header 3": "**EXAMPLE 6.5**",
"token_count": 297,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Predict the weight in pounds for an adult who is 5 feet, 11 inches tall. 1 cm = .3937 in; 1 lb = $0.454 \text{ kg.}^8$
E
(G
<sup>&</sup>lt;sup>7</sup>The equation of the line is weight = -57.738 + 0.839(height), where height is in centimeters and weight is in kilograms. <sup>8</sup>5 feet, 11 inches equals 71/.3... | {
"Header 1": "**6.2 Estimating a regression line using least squares**",
"Header 3": "**GUIDED PRACTICE 6.7**",
"token_count": 209,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
A least squares regression line functions as a statistical model that can be used to estimate the relationship between an explanatory and response variable. While the calculations for constructing a regression line are relatively simple, interpreting the linear model is not always straightforward. In addition to discus... | {
"Header 1": "**6.3 Interpreting a linear model**",
"token_count": 589,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
There are a variety of residual plots used to check the fit of a least squares line. The plots shown in this text are scatterplots in which the residuals are plotted on the vertical axis against predicted values from the model on the horizontal axis. Other residual plots may instead show values of the explanatory varia... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 3": "Examining patterns in residuals",
"token_count": 487,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The correlation coefficient r measures the strength of the linear relationship between two variables. However, it is more common to measure the strength of a linear fit using $r^2$ , which is commonly written as $R^2$ in the context of regression.<sup>12</sup>
The quantity $R^2$ describes the amount of variation... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 2": "**6.3.2** Using $R^2$ to describe the strength of a fit",
"token_count": 575,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
If a linear model has a very strong negative relationship with a correlation of -0.97, how much of the variation in the response is explained by the explanatory variable?<sup>14</sup>
<sup>3</sup>About 16.8%:
$$\frac{s_{\text{weight}}^2 - s_{\text{residuals}}^2}{s_{\text{weight}}^2} = \frac{442.53 - 368.1}{442.53} = ... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 2": "**6.3.2** Using $R^2$ to describe the strength of a fit",
"Header 3": "**GUIDED PRACTICE 6.11**",
"token_count": 222,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Although the response variable in linear regression is necessarily numerical, the predictor variable may be either numerical or categorical. This section explores the association between a country's infant mortality rate and whether or not 50% of the population has access to adequate sanitation facilities.
The World ... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 2": "**6.3.2** Using $R^2$ to describe the strength of a fit",
"Header 3": "**6.3.3 Categorical predictors with two levels**",
"token_count": 924,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Using the model in Equation [6.12,](#page-303-2) the prediction equation can be written
log(inf.mortality ) = 4*.*<sup>018</sup> <sup>−</sup> <sup>1</sup>*.*681(sanit.access)*.*
Exponentiating both sides of the equation yields
inf.mortality <sup>=</sup> *<sup>e</sup>* 4*.*018−1*.*681(sanit.access) *.*
When sani... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 2": "**6.3.2** Using $R^2$ to describe the strength of a fit",
"Header 3": "6.3. INTERPRETING A LINEAR MODEL 305",
"token_count": 322,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Depending on their position, data points in a scatterplot have varying degrees of contribution to the estimated parameters of a regression line. Points that are at particularly low or high values of the predictor (*x*) variable are said to have high leverage, and have a large influence on the estimated intercept and sl... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 2": "**6.3.2** Using $R^2$ to describe the strength of a fit",
"Header 3": "**6.3.4 Outliers in regression**",
"token_count": 773,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Once the influential DC point is removed, assess whether it is appropriate to use linear regression on these data by checking the four assumptions behind least squares regression: linearity, constant variability, independent observations, and approximate normality of the residuals. Refer to the residual plots shown in ... | {
"Header 1": "**6.3 Interpreting a linear model**",
"Header 2": "**6.3.2** Using $R^2$ to describe the strength of a fit",
"Header 3": "**GUIDED PRACTICE 6.14**",
"token_count": 462,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The previous sections in this chapter have focused on linear regression as a tool for summarizing trends in data and making predictions. These numerical summaries are analogous to the methods discussed in Chapter [1](#page-9-0) for displaying and summarizing data. Regression is also used to make inferences about a popu... | {
"Header 1": "**6.4 Statistical inference with regression**",
"token_count": 1039,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Typically, hypothesis testing in regression involves tests of whether the *x* and *y* variables are associated; in other words, whether the slope is significantly different from 0. In these settings, the null hypothesis is that there is no association between the explanatory and response variables, or $H_0: \beta_1 = ... | {
"Header 1": "**6.4 Statistical inference with regression**",
"Header 3": "6.4. STATISTICAL INFERENCE WITH REGRESSION",
"token_count": 295,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Statistical software is typically used to obtain *t*-statistics and *p*-values for inference with regression, since using the formulas for calculating standard error can be cumbersome.
The standard errors of $b_0$ and $b_1$ used in confidence intervals and hypothesis tests replace $\sigma$ with *s*, the standar... | {
"Header 1": "**6.4 Statistical inference with regression**",
"Header 3": "Formulas for calculating standard errors",
"token_count": 253,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Is there evidence of a significant association between number of doctors per 100,000 members of the population in a state and infant mortality rate? The numerical output that B returns is shown in Figure 6 16 $^{23}$
The numerical output that R returns is shown in Figure 6.16.<sup>23</sup>
The question implies tha... | {
"Header 1": "**6.4 Statistical inference with regression**",
"Header 3": "EXAMPLE 6.17",
"token_count": 688,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Calculate a 95% two-sided confidence interval for the slope parameter $\beta_1$ in the state-level infant mortality data.<sup>25</sup>
$<sup>^{23}</sup>$ Other software packages, such as Stata or Minitab, provide similar information but with slightly different labeling. $^{24}$ Calculations of the $R^2$ value ... | {
"Header 1": "**6.4 Statistical inference with regression**",
"Header 3": "**GUIDED PRACTICE 6.18**",
"token_count": 211,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Conducting a regression analysis with a numerical response variable and a categorical predictor with two levels is analogous to conducting a two-group hypothesis test.
For example, Section [6.3.3](#page-302-4) shows a regression model that compares the average infant mortality rate in countries with low access to san... | {
"Header 1": "**6.4 Statistical inference with regression**",
"Header 3": "Connection to two-group hypothesis testing",
"token_count": 960,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
As initially discussed in Section 6.2, the estimated regression line for the association between RFFT score and age from the 500 individuals in prevend.samp is
$$RFFT = 137.55 - 1.26(age).$$
Figure 6.19 shows the summary output from R when the regression model is fit. R also provides the value of $R^2$ as 0.285 a... | {
"Header 1": "6.5 Interval estimates with regression",
"Header 3": "6.5.1 Confidence intervals",
"token_count": 795,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The confidence interval for $E(Y|x^*)$ is computed using the standard error of the estimated mean of the regression model at a value of the predictor:
s.e.
$$(\widehat{E(Y|x^*)}) = \sqrt{s^2 \left(\frac{1}{n} + \frac{(x^* - \overline{x})^2}{\sum (x_i - \overline{x})^2}\right)} = s \sqrt{\frac{1}{n} + \frac{(x^* - \... | {
"Header 1": "6.5 Interval estimates with regression",
"Header 3": "6.5. INTERVAL ESTIMATES WITH REGRESSION",
"token_count": 913,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
After fitting a regression line, a **prediction interval** is used to estimate a range of values for a new observation of the response variable *Y* with predictor value $x^*$ ; that is, an observation not in the data used to estimate the line. The point estimate $\widehat{Y|x^*} = b_0 + b_1 x^*$ is the same as $\wi... | {
"Header 1": "6.5 Interval estimates with regression",
"Header 3": "6.5.2 Prediction intervals",
"token_count": 1255,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
This chapter provides only an introduction to simple linear regression; the next chapter, Chapter [7,](#page-329-0) expands on the principles of simple regression to models with more than one predictor variable.
When fitting a simple regression, be sure to visually assess whether the model is appropriate. Nonlinear t... | {
"Header 1": "**6.6 Notes**",
"token_count": 736,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
**[6.5](#page-451-8) Over-under, Part I.** Suppose we fit a regression line to predict the shelf life of an apple based on its weight. For a particular apple, we predict the shelf life to be 4.6 days. The apple's residual is -0.6 days. Did we over or under estimate the shelf-life of the apple? Explain your reasoning. ... | {
"Header 1": "6.7 Exercises",
"Header 2": "6.7.1 Examining scatterplots",
"Header 3": "6.7. EXERCISES 319",
"token_count": 1053,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
6.25 Husbands and wives. Part I. The scatterplot below summarizes husbands' and wives' heights in a random sample of 170 married couples in Britain, where both partners' ages are below 65 years. Summary output of the least squares fit for predicting wife's height from husband's height is also provided in the table.
!... | {
"Header 1": "6.7 Exercises",
"Header 2": "6.7.1 Examining scatterplots",
"Header 3": "6.7. EXERCISES",
"token_count": 1419,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
| | Estimate | Std. Error | t value | Pr(> t ) |
|-------------|----------|------------|---------|----------|
| (Intercept) | 137.55 | 5.02 | 27.42 | 0.000 |
| Age | -1.26 | 0.09 | -14.09 | 0.000 |
| | | | | df = 498 |
**[6.29](#pag... | {
"Header 1": "6.7 Exercises",
"Header 2": "6.7.1 Examining scatterplots",
"Header 3": "6.7. EXERCISES 329",
"token_count": 351,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
- **[7.1](#page-331-0) [Introduction to multiple linear regression](#page-331-0)**
- **[7.2](#page-333-0) [Simple versus multiple regression](#page-333-0)**
- **[7.3](#page-337-0) [Evaluating the fit of a multiple regression model](#page-337-0)**
- **[7.4](#page-341-0) [The general multiple linear regression model](#pa... | {
"Header 1": "**Multiple linear regression**",
"token_count": 690,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Statins are a class of drugs widely used to lower cholesterol. There are two main types of cholesterol: low density lipoprotein (LDL) and high density lipoprotein (HDL).[1](#page-331-1) Research suggests that adults with elevated LDL may be at risk for adverse cardiovascular events such as a heart attack or stroke. In ... | {
"Header 1": "**7.1 Introduction to multiple linear regression**",
"token_count": 1271,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
A simple linear regression model can be fit for an initial examination of the association between statin use and RFFT score,
$$E(\text{RFFT}) = \beta_0 + \beta_{\text{Statin}}(\text{Statin}).$$
RFFT scores in prevend.samp are approximately normally distributed, ranging between approximately 10 and 140, with no obvi... | {
"Header 1": "**7.2 Simple versus multiple regression**",
"token_count": 968,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Using the parameter estimates in Figure [7.5,](#page-334-1) write the prediction equation for the linear model. How does the predicted RFFT score for a 67-year-old not using statins compare to that of an individual of the same age who does use statins?
The equation of the linear model is
RFFT = 137 *.*8822 + 0*.*85... | {
"Header 1": "**7.2 Simple versus multiple regression**",
"Header 3": "**EXAMPLE 7.1**",
"token_count": 371,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Suppose two individuals are both taking statins; one individual is 50 years of age, while the other is 60 years of age. Compare their predicted RFFT scores.
From the model equation, the coefficient of age *β*Age is -1.2710; an increase in one unit of age (i.e., one year) is associated with a decrease in RFFT score of... | {
"Header 1": "**7.2 Simple versus multiple regression**",
"Header 3": "**EXAMPLE 7.2**",
"token_count": 401,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
What does the intercept represent in this model? Does the intercept have interpretive value?[9](#page-336-0)
As in simple linear regression, *t*-statistics can be used to test hypotheses about the slope coefficients; for this model, the two null hypotheses are *H*<sup>0</sup> : *β*Statin = 0 and *H*<sup>0</sup> : *β*... | {
"Header 1": "**7.2 Simple versus multiple regression**",
"Header 3": "**GUIDED PRACTICE 7.3**",
"token_count": 585,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The assumptions behind multiple regression are essentially the same as the four assumptions listed in Section 6.1 for simple linear regression. The assumption of linearity is extended to multiple regression by assuming that when only one predictor variable changes, it is linearly related to the change in the response v... | {
"Header 1": "7.3 Evaluating the fit of a multiple regression model",
"Header 3": "7.3.1 Using residuals to check model assumptions",
"token_count": 449,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Section 1.7 featured a case study examining the evidence for ethnic discrimination in the amount of financial support offered by the State of California to individuals with developmental disabilities. Although an initial look at the data suggested an association between expenditures and ethnicity, further analysis sugg... | {
"Header 1": "7.3 Evaluating the fit of a multiple regression model",
"Header 3": "EXAMPLE 7.4",
"token_count": 488,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Section [6.3.2](#page-301-3) provided two definitions of the *R* 2 statistic—it is the square of the correlation coefficient *r* between a response and the single predictor in simple linear regression, and equivalently, it is the proportion of the variation in the response variable explained by the model. In statistica... | {
"Header 1": "7.3 Evaluating the fit of a multiple regression model",
"Header 3": "**7.3.2 Using** *R* <sup>2</sup> **and adjusted** *R* <sup>2</sup> **with multiple regression**",
"token_count": 881,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
For multiple regression, the data consist of a response variable *Y* and *p* explanatory variables *X*1*,X*2*,...,Xp*. Instead of the simple regression model
$$Y = \beta_0 + \beta_1 X + \varepsilon,$$
multiple regression has the form
$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \dots + \beta_p X_p + ... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "**7.4.1 Model parameters and least squares estimation**",
"token_count": 408,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Figure [7.9](#page-342-0) shows an estimated regression model for RFFT with predictors Statin and Gender, where Gender is coded 0 for males and 1 for females.[11](#page-0-0) Based on the model, what are the estimated mean RFFT scores for the four groups defined by these two categorical predictors?[12](#page-342-1)
| ... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "**GUIDED PRACTICE 7.5**",
"token_count": 1103,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The test of the null hypothesis *H*<sup>0</sup> : *β<sup>k</sup>* = 0 is a test of whether the predictor *X<sup>k</sup>* is associated with the response variable. When a coefficient of a predictor equals 0, the predicted value of the response does not change when the predictor changes; i.e., a value of 0 indicates ther... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "Using *t*-tests for individual coefficients",
"token_count": 464,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
A test of the two-sided hypothesis
$$H_0: \beta_k = 0$$
vs. $H_A: \beta_k \neq 0$
is rejected with significance level *α* when
$$\frac{|b_k|}{\text{s.e.}(b_k)} > t_{\text{df}}^\star,$$
where *t ?* df is the point on a *<sup>t</sup>*-distribution with *<sup>n</sup>*−*p*−1 degrees of freedom and area (1−*α/*2) i... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "**TESTING A HYPOTHESIS ABOUT A REGRESSION COEFFICIENT**",
"token_count": 268,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
When all the model coefficients are 0, the predictors in the model, considered as a group, are not associated with the response; i.e., the response variable is not associated with any linear combination of the predictors. The *F*-statistic is used to test this null hypothesis of no association, using the following idea... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "The F-statistic for an overall test of the model",
"token_count": 447,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The *F*-statistic in regression is used to test the null hypothesis
$$H_0: \beta_1 = \beta_2 = \dots = \beta_p = 0$$
against the alternative that at least one of the coefficients is not 0.
Under the null hypothesis, the sampling distribution of the *F*-statistic is an *F*-distribution with parameters (*p,n* − *p*... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "**THE** *F***-STATISTIC IN REGRESSION**",
"token_count": 307,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The confidence and prediction intervals discussed in Section [6.5](#page-311-0) can be extended to multiple regression. Predictions based on specific values of the predictors are made by evaluating the estimated model at those values, and both confidence intervals for the mean and prediction intervals for a new observa... | {
"Header 1": "**7.4 The general multiple linear regression model**",
"Header 3": "Confidence and Prediction Intervals",
"token_count": 317,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
In the initial model fit with the PREVEND data, the variable Statin is coded 0 if the participant was not using statins, and coded 1 if the participant was a statin user. The category coded 0 is referred to as the reference category; in this model, statin non-users (Statin = 0) are the reference category. The estimated... | {
"Header 1": "**7.5 Categorical predictors with several levels**",
"token_count": 760,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Is RFFT score associated with educational level? Interpret the coefficients from the following model. Figure [7.12](#page-348-0) provides the R output for the regression model of RFFT versus educational level in prevend.samp. The variable Education has been converted to Education.factor, which has levels Primary, Lower... | {
"Header 1": "**7.5 Categorical predictors with several levels**",
"Header 3": "**EXAMPLE 7.6**",
"token_count": 768,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Suppose that the model for predicting RFFT score from educational level is fitted with Education, using the original numerical coding with 0, 1, 2, and 3; the R output is shown in Figure [7.13.](#page-348-1) What does this model imply about the change in mean RFFT between groups? Explain why this model is flawed.
Acc... | {
"Header 1": "**7.5 Categorical predictors with several levels**",
"Header 3": "**EXAMPLE 7.7**",
"token_count": 504,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
The earlier models fit to examine the association between cognitive ability and statin use showed that considering statin use alone could be misleading. While older participants tended to have lower RFFT scores, they were also more likely to be taking statins. Age was found to be a confounder in this setting—is it the ... | {
"Header 1": "**7.6 Reanalyzing the PREVEND data**",
"token_count": 914,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Figure 7.15 shows a plot of residuals vs predicted RFFT scores from the model in Figure 7.14 and a normal probability plot of the residuals. These plots show that the model fits the data reasonably well. The residuals show a slight increase in variability for larger predicted values, and the normal probability plot sho... | {
"Header 1": "**7.6 Reanalyzing the PREVEND data**",
"Header 3": "7.6. REANALYZING THE PREVEND DATA",
"token_count": 401,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
An important assumption in the multiple regression model
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_p x_p + \varepsilon$$
is that when one of the predictor variables *x<sup>j</sup>* changes by 1 unit and none of the other variables change, the predicted response changes by *β<sup>j</sup>* , regardles... | {
"Header 1": "**7.7 Interaction in regression**",
"token_count": 614,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
Using the output in Figure 7.16, write two separate model equations: one for diabetic individuals and one for non-diabetic individuals. Compare the two models.
For non-diabetics (Diabetes = $\emptyset$ ), the linear relationship between average cholesterol and age is $\widehat{\text{TotChol}} = 4.80 + 0.0075(\text{... | {
"Header 1": "**7.7 Interaction in regression**",
"Header 3": "**EXAMPLE 7.10**",
"token_count": 1336,
"source_pdf": "datasets/websources/Med_v1/med_textbook/biostat.pdf"
} |
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