106 Questions
What is the primary objective of statistical inference?
To test suitable hypotheses using data
What type of reasoning involves proceeding from the general to the specific?
Deductive reasoning
What is the purpose of deductive reasoning in science?
To test predictions
What does inductive reasoning involve?
Drawing conclusions from data
What happens if the data are inconsistent with the predictions from a hypothesis?
The hypothesis is rejected or modified
What is the limitation of deductive reasoning in hypothesis testing?
It cannot prove a hypothesis true
What do clinicians often use to determine the values of variables in a clinical situation?
Deductive reasoning and formulas
What is the difference between inductive and deductive reasoning?
Inductive is from specific to general, deductive is from general to specific
What type of error occurs when an investigator asserts that the data support a hypothesis, when in fact the hypothesis is false?
False-positive error
Why do investigators historically avoid false-positive errors?
Because of the principle 'first, do no harm'
What is the purpose of stating a null hypothesis?
To state that there is no real difference between groups
What happens if the data are not consistent with a hypothesis?
The null hypothesis is rejected
What is the difference between a one-tailed and a two-tailed hypothesis test?
A one-tailed test specifies a directional inclination, while a two-tailed test does not
What is the null hypothesis in a clinical trial of a drug designed to reduce high blood pressure?
There is no true difference between the treatment and control groups
What is the consequence of a false-negative error in a diagnostic test?
A patient may not receive necessary treatment
Why are high standards for the avoidance of type I error particularly important in medical practice?
To avoid harming patients
What is the consequence of a false-negative error in the study of a medical intervention?
A patient may not receive a potentially effective treatment
What is the alternative hypothesis in a clinical trial of a drug designed to reduce high blood pressure?
There is a true difference between the treatment and control groups
What is the primary difference between deductive and inductive reasoning?
Deductive reasoning proceeds from the general to the specific, while inductive reasoning seeks to find general principles from data
What is the purpose of statistical analysis in a study?
To find the general relationship between variables from specific data
What is the main difference between the approach of mathematics and statistics to the same equation?
Mathematics starts with a formula, while statistics starts with data
What is a hypothesis in the context of statistical testing?
A prediction about the outcome of a study
What is the primary goal of statistical tests of significance?
To determine the probability that an observed difference represents a true difference
What is the purpose of the p value in statistical testing?
To determine the probability of a false-positive conclusion
What is the consequence of a false-positive conclusion in statistical testing?
The retention of a false hypothesis
What is the basis for developing hypotheses in scientific research?
Previous experience
What is the ultimate goal of scientific research, according to the principles outlined?
To develop theories that are consistent with the data
What is the purpose of establishing an alpha level in hypothesis testing?
To set the maximum risk of making a false-positive error
What is the typical value of alpha commonly used in hypothesis testing?
p = 0.05
What is the purpose of performing a statistical test of significance?
To obtain the p-value and determine statistical significance
What does the p-value represent in hypothesis testing?
The probability of obtaining the observed result by chance
What is the usual criterion for rejecting the null hypothesis?
p-value ≤ alpha
What is the advantage of setting the alpha level before collecting data?
It avoids post hoc bias
What is the purpose of comparing the p-value with the alpha level?
To decide whether to reject or fail to reject the null hypothesis
What is a one-tailed test commonly used for?
Testing a directional hypothesis
What is the norm in hypothesis testing?
Two-tailed tests
What is the purpose of providing explanations for choosing a different alpha level?
To justify the deviation from the customary alpha level
Why do researchers reject the null hypothesis if the p-value is less than or equal to the alpha level?
Because the difference is unlikely to be due to chance alone
What is the purpose of testing for significance in medical research?
To avoid the use of ineffective therapies
What is the standard error used to estimate in a study?
The reliability of the findings
Why do researchers compare the means of two groups instead of just inspecting the means?
To determine if the observed difference is statistically significant
What is the primary difference between a standard deviation and a standard error?
A standard deviation is used for individual observations, while a standard error is used for means
What is the purpose of determining the average change in the treatment and control groups in a study?
To pursue tests to determine whether the difference was large enough to be unlikely to have occurred by chance alone
Why do researchers use statistical tests of significance in medical research?
To determine if the observed difference is real or due to chance
What is the assumption in a statistical test of significance?
That there is no difference between the groups
What is the primary difference between a null hypothesis and an alternative hypothesis?
A null hypothesis assumes no difference, while an alternative hypothesis assumes a difference
What is the primary use of confidence intervals?
To determine the significance of a mean or proportion
What does a 95% confidence interval represent?
The probability that the true effect of an intervention lies within a specific range
What is the implication of a confidence interval that includes the value of 1.0 for a risk ratio?
The risk ratio is not statistically significant
What is the advantage of a confidence interval over a p value?
It provides a range of values within which the true effect is likely to lie
What does a narrow confidence interval suggest?
The true effect of an intervention lies within a small range of values
What is the purpose of a confidence interval in hypothesis testing?
To provide a range of values within which the true effect is likely to lie
What is the relationship between a confidence interval and statistical significance?
A confidence interval provides a range of values that is related to statistical significance
What is the implication of a wide confidence interval?
The true effect of an intervention lies within a wide range of values
What is the main purpose of the standard error?
To estimate the probable error of the sample mean's estimate of the true population mean
What is the difference between the standard deviation and the standard error?
The standard deviation is used to estimate the variability of individual observations, while the standard error is used to estimate the variability of means
What is the purpose of reporting the sample size in the medical literature?
To enable the reader to convert the standard deviation to the standard error, if necessary
What is the 95% confidence interval used to estimate?
The range of values in which 95% of the means of repeated samples of the same size would be expected to fall
What is the main difference between the mean ± 1SD and the mean ± 1SE?
The mean ± 1SD is used to estimate the variability of individual observations, while the mean ± 1SE is used to estimate the variability of means
Why is it important to examine the reported data carefully in the medical literature?
To determine whether the SD or the SE is shown
What is the relationship between the sample size and the standard error?
The larger the sample size, the smaller the standard error
What is the purpose of the standard error in statistics?
To estimate the probable error of the sample mean's estimate of the true population mean
What is the distribution of the sample means?
A normal distribution
What is the primary function of critical ratios in tests of statistical significance?
To compare the observed ratio with the values in statistical tables
What is the formula for calculating a critical ratio in a test of statistical significance?
Parameter / Standard error of that parameter
What is the purpose of looking up the critical ratio in statistical tables?
To calculate the p value of a test
What is the relationship between the critical ratio and the standard error of a parameter?
The critical ratio is equal to the parameter divided by the standard error
What is the purpose of statistical tables in tests of statistical significance?
To look up the corresponding p value of a critical ratio
What is the main difference between the various tests of statistical significance?
The types of parameters compared
What is the common purpose of the various tests of statistical significance?
To compare two parameters and determine statistical significance
What is the primary goal of using tests of statistical significance in medical research?
To compare two parameters and determine statistical significance
What is the purpose of the t-test in research?
To test the null hypothesis
What is the purpose of the degrees of freedom (df) in the t-test?
To refer to the correct line in the t-distribution table
What is the null hypothesis in a t-test?
μ1 = μ2
What is the purpose of the sP2 in the t-test formula?
To estimate the population variance
What is the assumption in a two-sample t-test?
The samples are independent
What is the purpose of the numerator in the t-test formula?
To calculate the difference between the sample means
What does the t-test help investigators distinguish between?
Explained variation from random error
What is the p value in statistical analysis?
The probability of being in error if the null hypothesis of no difference is rejected
When is a one-tailed test commonly used?
When the new treatment is known to cost much more than the current therapy
What is the significance of a p value ≤ 0.05?
The difference is statistically significant
What is the purpose of the t-test in medical studies?
To distinguish between explained variation and unexplained variation
What is the difference between a one-tailed and a two-tailed test?
The alpha level is equally divided in a two-tailed test
What is the consequence of a false-positive error in statistical testing?
A false difference is detected
What is the purpose of statistical analysis in a study?
To distinguish between explained variation and unexplained variation
What is the significance of a two-tailed test?
It is used to detect differences in both directions
Why do many investigators dislike one-tailed tests?
Because they do not document if an intervention is significantly worse than the standard therapy
What is the main advantage of a paired t-test compared to Student's t-test?
It considers the variation from only one group of people
What is the formula for calculating the paired t-test?
t = (d - 0) / (s / sqrt(N))
What is the null hypothesis in a paired t-test?
The observed difference is equal to 0
What is the purpose of the paired t-test?
To determine if there is a change in the value of a continuous variable over time
What is the advantage of using a paired t-test over a two-sample t-test?
It is more robust because it considers the variation from only one group of people
What is the implication of a large value of t?
The p-value will be small
What is the sum of the squared deviations in a dataset commonly referred to as?
Total Sum of Squares (TSS)
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?
To explain the total sum of squares
What happens to the within-groups variation if other independent variables are added?
It decreases
What is the purpose of a z-test for proportions?
To compare differences between 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?
100%
What is the difference between the mean height for women and the mean height for men?
The between-groups variation
What is the purpose of calculating the z-statistic in a z-test for proportions?
To determine the p-value
What is the null hypothesis in a z-test for proportions?
The proportions are not significantly different
What is the purpose of calculating the standard error of the difference between proportions?
To calculate the z-statistic
What is the advantage of using a z-test for proportions over a t-test?
The z-test is specifically designed for proportions, unlike the t-test which is for means
What is the purpose of interpreting the result of a z-test for proportions?
To decide whether to reject the null hypothesis
What is the assumption underlying a z-test for proportions?
The data follow a binomial distribution
What is the purpose of calculating the confidence interval in a z-test for proportions?
To provide a range of values within which the true proportion is likely to lie
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.
Learn about the concept of statistical inference, its differences with deductive and inductive reasoning, and how to draw conclusions from data. This quiz covers the basics of statistical inference and its applications.
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