Updated: 5/22/2020

Statistic Definitions

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Introduction
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 pos
    disease neg
    test pos
    true positive
    a (6)
    false positive
    b
    test neg
    false negative
    c
      (4)
    true negative
    d
    TOTAL
    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 pos
        disease neg
        test pos
        true positive
        a
        false positive
        b (5)
        test neg
        false negative
        c
        true negative
        d (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 pos
      disease neg
      test pos
      true positive
      a
      false positive
      b
      test neg
      false negative
      c
      true negative
      d

False Negative Rate
  • Definition 
    • patients with disease who have a negative test result
  • Equation 
    • false negative rate = c / (a + c)

       
      disease pos
      disease neg
      test pos
      true positive
      a
      false positive
      b
      test neg
      false negative
      c
      true negative
      d
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 
      • use sensitivity, specificity, and prevalence to calculate the quadrants

         
        disease pos
        disease neg
        test pos
        true positive
        a (9)
        false positive
        b (4.5)
        test neg
        false negative
        c (1)
        true negative
        d (85.5)
        TOTAL
        a+c (10)
        b+d (90)
      • PPV = a / (a + b)
      • PPV = 9 / (9 + 4.5)
      • PPV = 67%
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?
    • solution
      • NPV = TN / (FN + TN)
      • NPV = 95 / (20 + 95)
      • NPV = 83%

         
        disease pos
        disease neg
        test pos
        true positive
        a (80)
        false positive
        b (5)
        test neg
        false negative
        c (20)
        true negative
        d (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
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
  • solution 
    • disease incidence in transfused
      • "YES" = 75 / (75 + 595) = .112
    • disease incidence in patients not transfued
      • "NO" = 16 / (16 + 712) = .022
    • relative risk (RR) = 0.112 / 0.022 = 5.09
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
  • Solution: 
    • OR = (75 x 712) / (595 x 16) = 5.61
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 
    • usually set at 95% by convention
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) 


        Choosing the Right Test
        Comparison Parametric Nonparametric
        Continous Data
        Two groups    
             Paired Dependent (paired) t-test Wilcoxon Rank-Sum 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 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
    • number of joints being
      followed and the number of failures are determined for
      each year after operation (number of joints being followed and the number of failures are determined for each year after operation each year of follow-up,  failure rate is calculated from the number of failures and the ‘number at risk’
    • 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. 

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