Pu 515 Applied Biostatistics Midterm Exam 1: Glucose Levels

Pu 515 Applied Biostatisticsmidterm Exam 1 Glucose Levels In Pa

Assess the statistical concepts related to the analysis of health-related data, including normal distribution calculations, probability, descriptive statistics, contingency tables, sensitivity and specificity of diagnostic tests, and relative risk measures. Address questions involving proportions, percentiles, hypothesis testing, and interpretation of statistical measures in medical research contexts.

Paper For Above instruction

Applied biostatistics plays a crucial role in understanding and interpreting health data, enabling researchers and clinicians to make informed decisions based on statistical evidence. This paper explores several key concepts in applied biostatistics through the lens of a set of practical questions, ranging from the distribution of glucose levels in non-diabetic patients to the evaluation of diagnostic tests and risk factors in various populations.

Analysis of Glucose Levels and Normal Distribution

Understanding the distribution of glucose levels in patients free of diabetes provides insights into what constitutes normal glucose physiology. Given that glucose levels are normally distributed with a mean of 120 mg/dL and a standard deviation of 16 mg/dL, the proportion of patients exceeding certain thresholds can be calculated using the properties of the normal distribution. For instance, the probability that a patient's glucose level exceeds 115 mg/dL involves computing the z-score: (115 - 120)/16 = -0.3125. Using standard normal tables or software, the area to the right of this z-score indicates that approximately 62.1% of patients have glucose levels above 115 mg/dL. Conversely, for a glucose level of 140 mg/dL, the z-score is (140 - 120)/16 = 1.25, corresponding to a percentile near the 89.4th percentile, indicating that a glucose level of 140 mg/dL exceeds 89.4% of observations.

Additionally, the probability that the mean glucose level in a sample of 12 patients exceeds 115 mg/dL can be assessed through the sampling distribution of the mean. The standard error is calculated as 16/√12 ≈ 4.6, and the z-score for the sample mean of 115 becomes (115 - 120)/4.6 ≈ -1.09. The associated probability is approximately 86%, indicating a high likelihood that the sample mean exceeds 115 mg/dL.

Descriptive Statistics of Body Mass Index (BMI)

Analyzing BMI data from 12 patients involves computing measures of central tendency and dispersion. The mean BMI provides an average measure, obtained by summing individual BMI scores and dividing by 12. The standard deviation quantifies variability around this mean, calculated via the square root of the variance, which involves the squared deviations from the mean. The median BMI, the middle value when data are ordered, offers a measure of central tendency less affected by skewness. Quartiles further describe data dispersion, with Q1 representing the 25th percentile and Q3 the 75th percentile. Outliers are identified by comparing data points beyond 1.5 times the interquartile range (Q3 - Q1), which helps determine whether extreme BMI scores deviate significantly from the overall distribution.

Analysis of Patient Classification and Probabilities

Contingency tables facilitate understanding the distribution of weight categories in relation to diabetes status. Calculating probabilities involves dividing specific subgroup counts by the total population. For example, the probability that a randomly selected patient is overweight is obtained by dividing the number of overweight patients by the total number of patients. The probability of a patient being obese and diabetic is calculated similarly, multiplying overall probabilities if independence assumptions are justified. The proportion of diabetics who are obese and the proportion of normal-weight patients who are not diabetic provide insights into disease risk factors and protective factors, respectively. Summing various categories yields overall proportions for weight classifications within the patient population, aiding in risk stratification and resource allocation.

Binomial Probabilities in Obese Patients

The probability that exactly half of 10 obese patients develop diabetes follows a binomial distribution with parameters n=10 and p=0.30. The probability mass function calculates this as C(10,5) (0.30)^5 (0.70)^5, which is approximately 0.266. The probability that none develop diabetes, with p=0.70 (no diabetes), is (0.70)^10 ≈ 0.028. The expected number of patients developing diabetes is n p = 10 0.30 = 3, representing the average outcome across many such samples.

Evaluation of a Screening Test's Diagnostic Performance

The sensitivity of a screening test measures its ability to correctly identify patients with impaired glucose tolerance. It is calculated as the number of true positives divided by the total number of actual positives, yielding 17 / (17 + 8) ≈ 67.7%. The false positive fraction indicates the proportion of patients without impaired glucose tolerance who test positive, calculated as false positives divided by all non-impaired patients: 13 / (13 + 37) ≈ 25.9%. These metrics are critical for evaluating the utility of diagnostic tools and guiding clinical decision-making.

Estimating Overweight and Obese Children from BMI Data

Children's BMI follows a normal distribution with a mean of 24.5 and a standard deviation of 6.2. The proportion of overweight children, with BMI between 25 and 30, is calculated by finding the probabilities at these bounds. The z-scores for 25 and 30 are (25 - 24.5)/6.2 ≈ 0.08 and (30 - 24.5)/6.2 ≈ 0.89, respectively. Using standard normal tables, the area between these z-scores indicates roughly 36.7% of children are overweight. For obesity (BMI ≥ 30), the z-score is 0.89, and the area to the right (greater than 0.89) corresponds to about 18.6%. The probability that the mean BMI of a random sample of 10 children exceeds 25 involves calculating the sampling distribution's standard error and the corresponding z-score, demonstrating the likelihood of observing a mean BMI above this threshold in practice.

Association Between Hypertension and Stroke

Data from a nationwide survey highlight the relationship between hypertension and stroke among individuals over 55. The incidence of stroke among hypertensive persons is 12 / (12 + 37) ≈ 24.5%, while among non-hypertensive persons, it is 4 / (4 + 26) ≈ 13.3%. The relative risk compares these two incidences: 0.245 / 0.133 ≈ 1.84, suggesting hypertensive individuals have nearly twice the risk of stroke. The odds ratio, a measure of association in case-control formats, is calculated as (12 26) / (4 37) ≈ 5.3, indicating a strong association between hypertension and stroke risk in this population.

True or False Statements

  • a) If there are outliers, then the mean will be greater than the median. False. Outliers can influence the mean significantly, but the median remains robust; the mean may be less than, greater than, or equal to the median depending on the data distribution.
  • b) The 90th percentile of the standard normal distribution is 1.645. True. This is a well-established critical value for normal distribution percentiles.
  • c) The mean is the 50th percentile of any normal distribution. True. By symmetry, the mean coincides with the median and mode in a normal distribution.
  • d) The mean is a better measure of location when there are no outliers. True. Without outliers, the mean provides a reliable central tendency measure.

Conclusion

Through various statistical analyses—ranging from probability calculations and descriptive statistics to risk assessments and diagnostic test evaluations—applied biostatistics enables effective interpretation of health data. Mastery of these concepts allows researchers and clinicians to identify patterns, evaluate risks, and improve diagnostic accuracy, ultimately contributing to better health outcomes and informed public health strategies.

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