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Updated: Jun 20 2024

Statistic Definitions

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• Introduction
• This topic covers a variety of statistical principles used in research and study design
• Measure of Central Tendency
• Mode
• defined as the value that occurs most often
• best for data which is allocated into distinct categories (nominal data)
• Median
• defined as the value that occurs at the middle of all values of the variable (half are greater, half are less)
• not affected by extreme values
• good for all levels of measurement except nominal data
• especially good for skewed distributions
• Mean
• defined as arithmetic average
• the most frequently used measure of central tendency
• uses all values of data
• highly sensitive to extreme values (especially skewed distributions)
• Sensitivity
• Definition
• probability that test results will be positive in patients with disease
• Equation
• sensitivity = a / (a + c) or
• sensitivity = TP / (TP + FN)
• Relevance
• sensitive tests are useful for screening since they are unlikely to miss a patient with disease
• Example
• a new test is developed to quickly diagnose HIV. There are 10 patients in the study group with the disease. Upon testing of all 10 patients, only 6 results return positive. What is the sensitivity of the new test?
• solution
• sensitivity = a / (a + c)
• sensitivity = 6 / 10
• sensitivity = 60%
•  Disease Positive Disease Negative Test Positive (a) true positive = 6 (b) false positive Test Negative (c) false negative = 4 (d) True negative TOTAL a + c = 10 b + d
• Specificity
• Definition
• probability test result will be negative in patients without disease
• Equation
• specificity= d / (b + d) or
• specificity = TN / (FP + TN)
• Relevance
• specific tests are useful for confirmation as they don't result in treatment of an unaffected individual
• Example
• in a population of 90 patients who are disease free, a test incorrectly diagnoses 5 patients with disease. What is the specificity of this test?
• solution
• specificity = d / (b + d)
• specificity = 85 / 90
• specificity = 94.4%  Disease Positive Disease Negative Test Positive (a) true positive (b) false positive = 5 Test Negative (c) false negative (d) true negative = 85 TOTAL a + c b + d (90)
• False Positive Rate
• Definition
• patients without the disease who have a positive test result
• Equation
• false positive rate = b / (b + d)  Disease Positive Disease Negative Test Positive (a) true positive (b) false positive Test Negative (c) false negative (d) true negative
• False Negative Rate
• Definition
• patients with disease who have a negative test result
• Equation
• false negative rate = c / (a + c)  Disease Positive Disease Negative Test Positive (a) true positive (b) false positive Test Negative (c) false negative (d) true negative
• Positive Predictive Value
• Definition
• probability patient with a positive test actually has the disease
• dependent on prevalence of disease
• Equation
• PPV = a / (a + b) or
• PPV = TP / (TP + FP)
• Example
• you are evaluating a new serum diagnostic test for Lyme disease that claims sensitivity 90% and specificity 0f 95%. The prevalence of Lyme disease is known to be 10% in late spring in the study of patients who present with fever, arthralgias, and rash.
• solution
• PPV = a / (a + b)
• PPV = 9 / (9 + 4.5)
• PPV = 67%
• use sensitivity, specificity, and prevalence to calculate the quadrants  Disease Positive Disease Negative Test Positive (a) true positive = 9 (b) false positive = 4.5 Test Negative (b) false negative = 1 (d) true negative = 85.5 TOTAL a+c = 10 b+d = 90
• Negative Predictive Value
• Definition
• probability patient with a negative test actually has no disease
• dependent on prevalence of disease
• Equation
• NPV = d / (c + d) or
• NPV = TN / (FN + TN)
• Example
• 200 patients are enrolled in a study to evaluate the accuracy of a ELISA-based test for the diagnosis of influenza. 100 patients were diagnosed by the gold-standard method. 80 of the patients with influenza had a positive ELISA-based test as did 5 of the patients without influenza. What is the negative predictive value of this test?
• solutionNPV = TN / (FN + TN)
• NPV = 95 / (20 + 95)
• NPV = 83%

 Disease Positive Disease Negative Test Positive (a) true positive = 80 (b) false positive = 5 Test Negative (c) false negative = 20 (d) true negative = 95
•  Likelihood Ratio
• Definition
• likelihood that a given test result would be expected in a patient with the target disorder compared to the likelihood that that same result would be expected in a patient without the target disorder
• Classification
• positive likelihood ratio
• definition
• describe how the likelihood of a disease is changed by a positive test result
• equation
• positive likelihood ratio = sensitivity / (1 - specificity)
• negative likelihood ratio
• definition
• describe how the likelihood of a disease is changed by a negative test result
• equation
• negative likelihood ratio = (1 - sensitivity) / specificity
• Incidence
• Definition
• number of newly reported cases of a disease in specific time period per unit measurement of population
• Prevalence
• Definition
• the total number of cases of a disease present in a location at any time point
• Determined by performing cross-sectional studies
• Relative Risk
• Definition
• risk of developing disease for people with known exposure compared to risk of developing disease without exposure
• obtained from cohort studies
• when RR > 1, the incidence of the outcome is greater in the exposed/treated group
• Equation
• incidence risk of YES = a / (a + b)
• incidence risk of NO =c / (c + d)
• relative risk = [(a / a + b)] / [(c / c + d)]  Disease Status Risk Present Absent Yes a b No c d
• Example
• a study is performed concerning the relationship between blood transfusions and the risk of developing hepatitis C. A group of patients is studied for three years.  Disease Status Transfused Hepatitis C Healthy Yes 75 595 No 16 712
• Odds Ratio
• Definition
• represents the odds that an outcome will occur given a particular exposure, compared to the odds that the outcome will occur without the exposure
• obtained from case-control studies (retrospective)
• also obtained from the output of logistic regression models
• odds ratio's approximate RR when the outcome is rare (usually defined as <10%)
• Equation
• OR = (a x d) / (b x c)  Disease Status Risk Present Absent Yes a b No c d
• Example
• a study is performed concerning the relationship between blood transfusions and the risk of developing hepatitis C. A group of patients is studied for three years.
•  Disease Status Transfused Hepatitis C Healthy Yes 75 595 No 16 712
• Number Needed to Treat
• Definition
• number of patients that must be treated in order to achieve one additional favorable outcome
• Equation
• number needed to treat = (1 / absolute risk reduction)
• Example
• you learn the number-needed-to-screen with FOBT is nearly 1000 to prevent colon cancer. What is the absolute risk reduction associated with FOBT?
• solution
• absolute risk reduction (ARR) = 1 / number needed to treat
• ARR = 1 / 1000
• ARR = .1%
• Post-test Odds of Disease
• Equations
• post-test probability = (pretest probability) X (likelihood ratio)
• likelihood ratio = sensitivity / (1 - specificity)
• pre-test odds = pre-test probability / (1 - pre-test probability)
• post-test probability = post-test odds / (post-test odds + 1)
• Power
• Definition
• an estimate of the probability a study will be able to detect a true effect of the intervention
• a power analysis to determine sample size should be performed prior to initiation of the study
• Equation
• power = 1 - (probability of a type-II, or beta error)
• Effect size
• Definition
• magnitude of the difference in the means of the control and experimental groups in a study with respect to the pooled standard deviation
• Variance
• Definition
• an estimate of the variability of each individual data point from the mean
• Type II Error (beta)
• Definition
• a false negative difference that can occur by
• detecting no difference when there is a difference or
• accepting a null hypothesis when it is false and should be rejected
• Equation
• power = 1 - (type-II error)
• Clinical significance
• a study that fails to find a difference may be because
• there actually is no difference or
• the study is not adequately powered
• Type I Error (alpha)
• Definition
• rejecting a null hypothesis even though it is true
• Clinical significance
• by definition, alpha-error rate is set to .05, meaning there is a 1/20 chance a type-I error has occurred
• Related principle
• Bonferroni correction
• post-hoc statistical correction made to P values when several dependent or independent statistical tests are being performed simultaneously on a single data set
• Confidence Interval
• Definition
• the interval that will include a specific parameter of interest, if the experiment is repeated
• 95% and 99% most commonly used
• 95% calculated based on mean +/- 1.96 standard deviations
• most commonly used by convention
• 99% calculated based on mean +/- 2.58 standard deviations
• Clinical significance
• Infers statistical significance, precision of findings, and clinical difference
• Statistical Inference
• Definition
• used to test specific hypotheses about associations or differences among groups of subjects/sample data
• Classification
• parametric inferential statistics
• continuous data that is normally distributed
• nonparametric inferential statistics
• continuous data that is not normally distributed (skewed)
• categorical data
• Study types
• when comparing two means
• Student's t-test
• used for parametric data
• Mann-Whitney or Wilcoxon rank sum test
• used for non-parametric data
• when comparing proportions
• chi-square test
• used for two or more groups of categorical data
• Fisher exact test
• used when sample sizes are small or
• number of occurrences in a group is low
• when comparing three or more groups
• Analysis of variance (ANOVA)
• one-way ANOVA for one independent variable and two-way ANOVA for two independent variables
• data must be normally distributed
•  Choosing the Right Test Comparison Parametric Nonparametric Continuous Data Two groups Paired Dependent (paired) t-test Wilcoxon Signed-Rank Test Unpaired Independent t-test Mann-Whitney U test Three or more groups Analysis of variance (ANOVA) Kruskal-Wallis test Categorical data Two or more variables Chi-square Chi-square Two or more variables (when the sample size is small) Fisher exact test Fisher exact test
• Funnel Plot
• Definition
• is a simple scatter plot of the intervention effect estimates from individual studies against some measure of each study’s size or precision and is used to detect publication bias in meta-analyses
• Clinical Significance
• this method is based on the fact that larger studies have smaller variability, whereas small studies, which are more numerous, have larger variability. Thus the plot of a sample of studies without publication bias will produce a symmetrical, inverted-funnel-shaped scatter, whereas a biased sample will result in a skewed plot.
• Receiver Operating Characteristic (ROC) Curve
• Definition
• a graphical representation of the diagnostic ability of different tests
• used to determine responsiveness
• Variables
• False positive rate (1 - specificity)
• is plotted on the x-axis
• True positive rate (sensitivity)
• is plotted on the y-axis
• Interpretation
• Area under the ROC curve (C-statistic)
• used to compare different tests, higher C-statistics mean better diagnostic ability of test
• an area under the ROC curve of 0.5 is a useless test
• Survivorship Analysis
• Overview
• often used to measure success of joint replacements
• analyzes data from patients with different lengths of follow-up
• for analysis, it is assumed that all patients had their operation simultaneously
• chance of implant surviving for a particular length of time is calculated as the survival rate
• calculation method is either life table or product limit method
• May be analyzed with the Kaplan-Meier method
• Life table method
• annual success rate, determined from the failure rate, is cumulated to give a survival rate for each successive year, this can change only once per year
• Product limit method
• same as life table method, but the survival rate is recalculated each time a failure occurs
• Minimal Clinically Important Difference (MCID)
• The difference in outcome measures that will have clinical relevance
• Difficult to study and measure, very few outcome tools have established and universally accepted MCID
• Helps to reconcile the statistical significance and clinical relevance of study results that use outcome tools
• Repeated Measurement Reliability
• Interrater/interobserver Reliability
• A measurement of the degree of agreement between two or more assessors
• Measured with Cohen's kappa
• Intrarater/intraobserver Reliability
• A measurement of the reliability of a single assessor making multiple measurements/observations of a single subject
• Measured with Intraclass correlation coefficient (ICC)
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