Lesson 13.3: Biostatistics and Quantitative Reasoning

Introduction

In this lesson, we will explore biostatistics and quantitative reasoning, essential components of understanding health systems and research outcomes. Our objectives include:

Understanding concepts such as sensitivity, specificity, predictive values, and likelihood ratios.
Learning about measures of risk and association, hypothesis testing, and types of error.
Interpreting confidence intervals, p-values, and the results of various studies.
Calculating and interpreting test performance and risk measures.
Understanding statistical significance, different errors, and the implications of confidence intervals.

To hook your interest, consider this: every day, healthcare providers use tests to determine whether patients have diseases. Understanding how these tests function statistically can help us make better decisions about patient care.

Sensitivity, Specificity, Predictive Values, and Likelihood Ratios

Sensitivity

Sensitivity is the ability of a test to correctly identify individuals with a disease, expressed mathematically as:

$$\text{Sensitivity} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Negatives}}$$

For example, if a new test for a disease correctly identifies 90 out of 100 patients who have the disease, the sensitivity is:

$$\text{Sensitivity} = \frac{90}{90 + 10} = \frac{90}{100} = 0.90$$

This means the test has a sensitivity of 90%. A common misconception is to confuse sensitivity with the overall accuracy of a test. Sensitivity only reflects how well a test identifies true cases without considering those who do not have the condition.

Specificity

Specificity, on the other hand, measures the ability of a test to correctly identify individuals without the disease, given by:

$$\text{Specificity} = \frac{\text{True Negatives}}{\text{True Negatives} + \text{False Positives}}$$

Suppose a test correctly detects that 85 out of 100 healthy individuals do not have the disease, the specificity is calculated as:

$$\text{Specificity} = \frac{85}{85 + 15} = \frac{85}{100} = 0.85$$

This shows that the specificity is also 85%. Here, the misconception lies in thinking that a high specificity means that the test is inherently better. The balance between sensitivity and specificity is crucial, as they help determine how trustworthy a test is in both identifying and ruling out a condition.

Predictive Values

Predictive values include:

Positive Predictive Value (PPV): The probability that individuals with a positive test result truly have the disease:

$$\text{PPV} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Positives}}$$

Negative Predictive Value (NPV): The probability that individuals with a negative test result truly do not have the disease:

$$\text{NPV} = \frac{\text{True Negatives}}{\text{True Negatives} + \text{False Negatives}}$$

For instance, if a test has 70 true positives and 30 false positives, the PPV would be:

$$\text{PPV} = \frac{70}{70 + 30} = 0.70$$

Likelihood Ratios

Likelihood ratios combine the sensitivity and specificity of a test into a single measure:

Positive Likelihood Ratio (LR+):

$$LR+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}}$$

Negative Likelihood Ratio (LR-):

$$LR- = \frac{1 - \text{Sensitivity}}{\text{Specificity}}$$

In a case study where a test has a sensitivity of 90% and a specificity of 80%, the positive likelihood ratio is calculated as:

$$LR+ = \frac{0.90}{1 - 0.80} = \frac{0.90}{0.20} = 4.5$$

A likelihood ratio greater than 1 indicates that the test result is associated with a higher probability of the disease.

Measures of Risk and Association

Measures of Risk

Risk measures help analyze how exposure to a particular factor influences the incidence of a disease. Common measures include:

Risk Ratio (Relative Risk):

$$\text{Risk Ratio} = \frac{\text{Risk in Exposed Group}}{\text{Risk in Unexposed Group}}$$

In a study of smokers versus non-smokers, if 40 out of 100 smokers develop lung cancer (40% risk) and 10 out of 100 non-smokers develop lung cancer (10% risk), the risk ratio is calculated as:

$$\text{Risk Ratio} = \frac{0.40}{0.10} = 4$$

This indicates that smokers are four times more likely to develop lung cancer than non-smokers.

Odds Ratio:

$$\text{Odds Ratio} = \frac{\text{Odds of Exposure in Cases}}{\text{Odds of Exposure in Controls}}$$

Association Measures

The association between exposure and outcome can also be examined through:

Cohort Studies: Following a population to see who develops the outcome.
Case-Control Studies: Comparing those with the outcome to those without it.
Cross-Sectional Studies: Observing populations at a single point in time for both exposure and outcome.

Misinterpretations often arise around the strength of the relationship; correlation does not imply causation, and understanding the context is vital in clinical practice.

Hypothesis Testing and Error Types

Hypothesis Testing

Hypothesis testing is a core methodology in statistics where researchers define a null hypothesis (H0) and an alternative hypothesis (H1). The process involves:

Formulating hypotheses.
Selecting a significance level (usually $\alpha = 0.05$).
Conducting a statistical test to determine a p-value.

A p-value less than the significance level indicates that the null hypothesis can be rejected, suggesting that the result is statistically significant.

Types of Errors

There are two major types of errors in hypothesis testing:

Type I Error: Rejecting the null hypothesis when it is actually true (false positive).
Type II Error: Failing to reject the null hypothesis when it is false (false negative).

Understanding these errors helps clinicians assess the reliability of study results and the implications of their decisions based on these tests.

Interpreting Test Performance and Risk Measures

Interpretation of test performance involves contextual understanding. For instance, a p-value of 0.04 indicates that there is only a 4% chance of observing the data if the null hypothesis were true. This does not ensure that the alternative hypothesis is correct; it suggests statistical significance but requires clinical relevance.

Confidence intervals provide a range of values that likely contain the true population parameter, offering insight into the precision of the estimate. For instance, a 95% confidence interval for a mean difference shows where the true mean difference lies 95% of the time in repeated sampling.

Conclusion

In conclusion, biostatistics and quantitative reasoning are critical for evaluating health studies and making informed decisions in clinical practice. Understanding concepts such as sensitivity, specificity, predictive values, risk measures, and hypothesis testing enhances interpretation and application of research findings.

Study Notes

Sensitivity: True positives/(True positives + False negatives)
Specificity: True negatives/(True negatives + False positives)
Positive Predictive Value (PPV): True positives/(True positives + False positives)
Negative Predictive Value (NPV): True negatives/(True negatives + False negatives)
Risk Ratio: Risk in exposed/Risk in unexposed
P-value: Probability of seeing the observed data given the null hypothesis
Type I Error: False positive
Type II Error: False negative