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Questions and Answers
What is the primary objective of statistical inference?
What is the primary objective of statistical inference?
- To test suitable hypotheses using data (correct)
- To interpret data without inductive reasoning
- To arrange and describe data
- To draw conclusions from qualitative information
What type of reasoning involves proceeding from the general to the specific?
What type of reasoning involves proceeding from the general to the specific?
- Inductive reasoning
- Deductive reasoning (correct)
- Predictive reasoning
- Abductive reasoning
What is the purpose of deductive reasoning in science?
What is the purpose of deductive reasoning in science?
- To test predictions (correct)
- To interpret data
- To prove hypotheses true
- To form hypotheses
What does inductive reasoning involve?
What does inductive reasoning involve?
What happens if the data are inconsistent with the predictions from a hypothesis?
What happens if the data are inconsistent with the predictions from a hypothesis?
What is the limitation of deductive reasoning in hypothesis testing?
What is the limitation of deductive reasoning in hypothesis testing?
What do clinicians often use to determine the values of variables in a clinical situation?
What do clinicians often use to determine the values of variables in a clinical situation?
What is the difference between inductive and deductive reasoning?
What is the difference between inductive and deductive reasoning?
What type of error occurs when an investigator asserts that the data support a hypothesis, when in fact the hypothesis is false?
What type of error occurs when an investigator asserts that the data support a hypothesis, when in fact the hypothesis is false?
Why do investigators historically avoid false-positive errors?
Why do investigators historically avoid false-positive errors?
What is the purpose of stating a null hypothesis?
What is the purpose of stating a null hypothesis?
What happens if the data are not consistent with a hypothesis?
What happens if the data are not consistent with a hypothesis?
What is the difference between a one-tailed and a two-tailed hypothesis test?
What is the difference between a one-tailed and a two-tailed hypothesis test?
What is the null hypothesis in a clinical trial of a drug designed to reduce high blood pressure?
What is the null hypothesis in a clinical trial of a drug designed to reduce high blood pressure?
What is the consequence of a false-negative error in a diagnostic test?
What is the consequence of a false-negative error in a diagnostic test?
Why are high standards for the avoidance of type I error particularly important in medical practice?
Why are high standards for the avoidance of type I error particularly important in medical practice?
What is the consequence of a false-negative error in the study of a medical intervention?
What is the consequence of a false-negative error in the study of a medical intervention?
What is the alternative hypothesis in a clinical trial of a drug designed to reduce high blood pressure?
What is the alternative hypothesis in a clinical trial of a drug designed to reduce high blood pressure?
What is the primary difference between deductive and inductive reasoning?
What is the primary difference between deductive and inductive reasoning?
What is the purpose of statistical analysis in a study?
What is the purpose of statistical analysis in a study?
What is the main difference between the approach of mathematics and statistics to the same equation?
What is the main difference between the approach of mathematics and statistics to the same equation?
What is a hypothesis in the context of statistical testing?
What is a hypothesis in the context of statistical testing?
What is the primary goal of statistical tests of significance?
What is the primary goal of statistical tests of significance?
What is the purpose of the p value in statistical testing?
What is the purpose of the p value in statistical testing?
What is the consequence of a false-positive conclusion in statistical testing?
What is the consequence of a false-positive conclusion in statistical testing?
What is the basis for developing hypotheses in scientific research?
What is the basis for developing hypotheses in scientific research?
What is the ultimate goal of scientific research, according to the principles outlined?
What is the ultimate goal of scientific research, according to the principles outlined?
What is the purpose of establishing an alpha level in hypothesis testing?
What is the purpose of establishing an alpha level in hypothesis testing?
What is the typical value of alpha commonly used in hypothesis testing?
What is the typical value of alpha commonly used in hypothesis testing?
What is the purpose of performing a statistical test of significance?
What is the purpose of performing a statistical test of significance?
What does the p-value represent in hypothesis testing?
What does the p-value represent in hypothesis testing?
What is the usual criterion for rejecting the null hypothesis?
What is the usual criterion for rejecting the null hypothesis?
What is the advantage of setting the alpha level before collecting data?
What is the advantage of setting the alpha level before collecting data?
What is the purpose of comparing the p-value with the alpha level?
What is the purpose of comparing the p-value with the alpha level?
What is a one-tailed test commonly used for?
What is a one-tailed test commonly used for?
What is the norm in hypothesis testing?
What is the norm in hypothesis testing?
What is the purpose of providing explanations for choosing a different alpha level?
What is the purpose of providing explanations for choosing a different alpha level?
Why do researchers reject the null hypothesis if the p-value is less than or equal to the alpha level?
Why do researchers reject the null hypothesis if the p-value is less than or equal to the alpha level?
What is the purpose of testing for significance in medical research?
What is the purpose of testing for significance in medical research?
What is the standard error used to estimate in a study?
What is the standard error used to estimate in a study?
Why do researchers compare the means of two groups instead of just inspecting the means?
Why do researchers compare the means of two groups instead of just inspecting the means?
What is the primary difference between a standard deviation and a standard error?
What is the primary difference between a standard deviation and a standard error?
What is the purpose of determining the average change in the treatment and control groups in a study?
What is the purpose of determining the average change in the treatment and control groups in a study?
Why do researchers use statistical tests of significance in medical research?
Why do researchers use statistical tests of significance in medical research?
What is the assumption in a statistical test of significance?
What is the assumption in a statistical test of significance?
What is the primary difference between a null hypothesis and an alternative hypothesis?
What is the primary difference between a null hypothesis and an alternative hypothesis?
What is the primary use of confidence intervals?
What is the primary use of confidence intervals?
What does a 95% confidence interval represent?
What does a 95% confidence interval represent?
What is the implication of a confidence interval that includes the value of 1.0 for a risk ratio?
What is the implication of a confidence interval that includes the value of 1.0 for a risk ratio?
What is the advantage of a confidence interval over a p value?
What is the advantage of a confidence interval over a p value?
What does a narrow confidence interval suggest?
What does a narrow confidence interval suggest?
What is the purpose of a confidence interval in hypothesis testing?
What is the purpose of a confidence interval in hypothesis testing?
What is the relationship between a confidence interval and statistical significance?
What is the relationship between a confidence interval and statistical significance?
What is the implication of a wide confidence interval?
What is the implication of a wide confidence interval?
What is the main purpose of the standard error?
What is the main purpose of the standard error?
What is the difference between the standard deviation and the standard error?
What is the difference between the standard deviation and the standard error?
What is the purpose of reporting the sample size in the medical literature?
What is the purpose of reporting the sample size in the medical literature?
What is the 95% confidence interval used to estimate?
What is the 95% confidence interval used to estimate?
What is the main difference between the mean ± 1SD and the mean ± 1SE?
What is the main difference between the mean ± 1SD and the mean ± 1SE?
Why is it important to examine the reported data carefully in the medical literature?
Why is it important to examine the reported data carefully in the medical literature?
What is the relationship between the sample size and the standard error?
What is the relationship between the sample size and the standard error?
What is the purpose of the standard error in statistics?
What is the purpose of the standard error in statistics?
What is the distribution of the sample means?
What is the distribution of the sample means?
What is the primary function of critical ratios in tests of statistical significance?
What is the primary function of critical ratios in tests of statistical significance?
What is the formula for calculating a critical ratio in a test of statistical significance?
What is the formula for calculating a critical ratio in a test of statistical significance?
What is the purpose of looking up the critical ratio in statistical tables?
What is the purpose of looking up the critical ratio in statistical tables?
What is the relationship between the critical ratio and the standard error of a parameter?
What is the relationship between the critical ratio and the standard error of a parameter?
What is the purpose of statistical tables in tests of statistical significance?
What is the purpose of statistical tables in tests of statistical significance?
What is the main difference between the various tests of statistical significance?
What is the main difference between the various tests of statistical significance?
What is the common purpose of the various tests of statistical significance?
What is the common purpose of the various tests of statistical significance?
What is the primary goal of using tests of statistical significance in medical research?
What is the primary goal of using tests of statistical significance in medical research?
What is the purpose of the t-test in research?
What is the purpose of the t-test in research?
What is the purpose of the degrees of freedom (df) in the t-test?
What is the purpose of the degrees of freedom (df) in the t-test?
What is the null hypothesis in a t-test?
What is the null hypothesis in a t-test?
What is the purpose of the sP2 in the t-test formula?
What is the purpose of the sP2 in the t-test formula?
What is the assumption in a two-sample t-test?
What is the assumption in a two-sample t-test?
What is the purpose of the numerator in the t-test formula?
What is the purpose of the numerator in the t-test formula?
What does the t-test help investigators distinguish between?
What does the t-test help investigators distinguish between?
What is the p value in statistical analysis?
What is the p value in statistical analysis?
When is a one-tailed test commonly used?
When is a one-tailed test commonly used?
What is the significance of a p value ≤ 0.05?
What is the significance of a p value ≤ 0.05?
What is the purpose of the t-test in medical studies?
What is the purpose of the t-test in medical studies?
What is the difference between a one-tailed and a two-tailed test?
What is the difference between a one-tailed and a two-tailed test?
What is the consequence of a false-positive error in statistical testing?
What is the consequence of a false-positive error in statistical testing?
What is the purpose of statistical analysis in a study?
What is the purpose of statistical analysis in a study?
What is the significance of a two-tailed test?
What is the significance of a two-tailed test?
Why do many investigators dislike one-tailed tests?
Why do many investigators dislike one-tailed tests?
What is the main advantage of a paired t-test compared to Student's t-test?
What is the main advantage of a paired t-test compared to Student's t-test?
What is the formula for calculating the paired t-test?
What is the formula for calculating the paired t-test?
What is the null hypothesis in a paired t-test?
What is the null hypothesis in a paired t-test?
What is the purpose of the paired t-test?
What is the purpose of the paired t-test?
What is the advantage of using a paired t-test over a two-sample t-test?
What is the advantage of using a paired t-test over a two-sample t-test?
What is the implication of a large value of t?
What is the implication of a large value of t?
What is the sum of the squared deviations in a dataset commonly referred to as?
What is the sum of the squared deviations in a dataset commonly referred to as?
What is the purpose of seeking to determine how much of the variation is caused by gender and how much is caused by other factors?
What is the purpose of seeking to determine how much of the variation is caused by gender and how much is caused by other factors?
What happens to the within-groups variation if other independent variables are added?
What happens to the within-groups variation if other independent variables are added?
What is the purpose of a z-test for proportions?
What is the purpose of a z-test for proportions?
What is the percentage of variation that would be explained by gender if all women were of equal height, all men were of equal height, and men were taller than women?
What is the percentage of variation that would be explained by gender if all women were of equal height, all men were of equal height, and men were taller than women?
What is the difference between the mean height for women and the mean height for men?
What is the difference between the mean height for women and the mean height for men?
What is the purpose of calculating the z-statistic in a z-test for proportions?
What is the purpose of calculating the z-statistic in a z-test for proportions?
What is the null hypothesis in a z-test for proportions?
What is the null hypothesis in a z-test for proportions?
What is the purpose of calculating the standard error of the difference between proportions?
What is the purpose of calculating the standard error of the difference between proportions?
What is the advantage of using a z-test for proportions over a t-test?
What is the advantage of using a z-test for proportions over a t-test?
What is the purpose of interpreting the result of a z-test for proportions?
What is the purpose of interpreting the result of a z-test for proportions?
What is the assumption underlying a z-test for proportions?
What is the assumption underlying a z-test for proportions?
What is the purpose of calculating the confidence interval in a z-test for proportions?
What is the purpose of calculating the confidence interval in a z-test for proportions?
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Study Notes
Nature and Purpose of Statistical Inference
- Statistical inference is the process of drawing conclusions from data using statistical methods to describe and arrange data and test hypotheses.
- There are two types of reasoning: deductive and inductive.
- Deductive reasoning proceeds from the general to the specific, whereas inductive reasoning seeks to find generalizations and principles from data.
Differences between Mathematics and Statistics
- Mathematics and statistics approach the same basic equation (y = mx + b) in different ways.
- In mathematics, the constants (m and b) are known, and the variables (x and y) are unknown.
- In statistics, the variables (x and y) are known, and the constants (m and b) are unknown and need to be estimated.
Process of Testing Hypotheses
- Hypotheses are predictions about what the examination of collected data will show.
- The null hypothesis states that there is no real difference between the means (or proportions) of the groups being compared.
- The alternative hypothesis states that there is a true difference between the groups being compared.
- The five steps of hypothesis testing are:
- Develop the null and alternative hypotheses.
- Establish an appropriate alpha level (usually 0.05).
- Perform a suitable test of statistical significance on collected data.
- Compare the p-value from the test with the alpha level.
- Reject or fail to reject the null hypothesis.
Alpha Level and p-Value
- The alpha level is the highest risk of making a false-positive error that the investigator is willing to accept.
- The p-value is the probability of obtaining the observed result by chance rather than as a result of a true effect.
- If the p-value is ≤ alpha level, the null hypothesis is rejected, and the alternative hypothesis is accepted.
Types of Errors
- False-positive error (type I error): asserting that the data support a hypothesis when it is false.
- False-negative error (type II error): failing to assert that the data support a hypothesis when it is true.
Process of Testing a Null Hypothesis
- Develop the null and alternative hypotheses.
- Establish an appropriate alpha level.
- Perform a suitable test of statistical significance on collected data.
- Compare the p-value from the test with the alpha level.
- Reject or fail to reject the null hypothesis.
Variation in Individual Observations and in Multiple Samples
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Most tests of significance relate to a difference between two means or proportions of a variable.
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The standard error is an unbiased estimate of the standard error in the entire population from whom the sample was taken.
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The standard error enables investigators to estimate the probable amount of error around a quantitative assertion and to perform tests of statistical significance.### Standard Deviation (SD) and Standard Error (SE)
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The formula to convert SD to SE is: SE = SD / √N
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A larger sample size (N) results in a smaller standard error, providing a better estimate of the population mean
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The sample mean is often near the true mean, but can be farther away from the average of the sample means
Reporting Means
- Means are often reported as mean ± 1SD or mean ± 1SE in the medical literature
- SD or SE can be converted to each other if the sample size is known
- Journals may have a policy on which one to report, and the sample size should always be shown
Confidence Intervals
- SD shows the variability of individual observations, while SE shows the variability of means
- Mean ± 1.96 SD estimates the range in which 95% of individual observations would fall
- Mean ± 1.96 SE estimates the range in which 95% of the means of repeated samples would fall
- The 95% confidence interval can be calculated from the mean ± 1.96 SE and represents the range of values in which the true mean of the underlying population is likely to fall
Confidence Intervals as a Test
- Confidence intervals can be used to determine whether a mean or proportion differs significantly from a fixed value
- If the confidence interval includes the fixed value, it means the difference is not statistically significant
- A narrow confidence interval indicates a precise estimate of the true effect, while a wide interval suggests a wider range of possible values
Tests of Statistical Significance
- Tests of statistical significance allow investigators to compare two parameters (means or proportions) and determine if the difference between them is statistically significant.
- Types of tests:
- T-tests: compare differences between means (one-tailed or two-tailed Student's t-test, paired t-test)
- Z-tests: compare differences between proportions
Critical Ratios
- Critical ratios are used to obtain a p-value to make a decision on the null hypothesis.
- Formula: Critical ratio = Parameter / Standard Error (SE) of that parameter
- The value of the critical ratio is looked up in a statistical table to determine the corresponding p-value.
Standard Error (SE) of the Difference
- SE of the difference between means:
sP2 = (sE2 + sC2) / (NE + NC - 2) - SE of the difference between proportions:
SEp = sqrt[p(1-p) / N]
T-Tests
- One-tailed and two-tailed t-tests:
- One-tailed: tests if the difference is greater than or less than a certain value (usually 0)
- Two-tailed: tests if the difference is significantly different from 0
- Formula:
t = (x1 - x2 - 0) / (sP2 * [(1/NE) + (1/NC)]) - Degrees of freedom:
df = NE + NC - 2
Paired T-Test
- Used for "before and after" experiments, where individual patients serve as their own control.
- More robust than Student's t-test because it considers variation from only one group of people.
- Formula:
t = (d - 0) / (sd / sqrt(N))
Z-Tests
- Used to compare differences between proportions.
- Formula:
z = (p1 - p2 - 0) / sqrt[p(1-p) * ((1/N1) + (1/N2))] - Standard error of the proportion:
SEp = sqrt[p(1-p) / N]
Interpretation of Results
- If the value of t or z is large, the p-value is small, indicating a statistically significant difference.
- A p-value of ≤0.05 is typically considered statistically significant.
Special Considerations
- Variation between groups versus variation within groups:
- The difference between two groups is found to be statistically significant, but it is important to ask why the groups are different and how much of the total variation is explained by the variable defining the two groups.
- The biostatistician would seek to determine the amount of the total variation in height that is explained by the gender difference.
- Measuring the total sum of squares (TSS) and the sum of squares (SS) for men and women:
- TSS is the total amount of variation that needs to be explained in the data set.
- SS is the unexplained variation within each group.
- The difference between TSS and SS is the explained variation, which is the amount of variation explained by the variable gender.
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