Statistic Definitions

• This topic covers the following statistical principles   
  • Measures of Central Tendency
  • Sensitivity
  • Specificity
  • False Positive Rate 
  • False Negative Rate
  • Positive Predictive Value
  • Negative Predictive Value
  • Likelihood Ratio
  • Incidence
  • Prevalence
  • Relative Risk
  • Odds Ratio
  • Number Needed to Treat
  • Post-test Odds of Disease
  • Power
  • Effect Size
  • Variance
  • Type II (beta) Error
  • Type I (alpha) Error
  • Confidence Interval
  • Statistical Inference 
  • Funnel plot
  • Receiver Operating Characteristic (ROC) Curve
    
  • Survivorship Analysis
  • Minimal Clinically Important Difference (MCID)

• 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)

• 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

• 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)

• 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

    
    

• 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

• 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%
      

• 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

• Definition
  • number of newly reported cases of a disease in specific time period per unit measurement of population 

• Definition
  • the total number of cases of a disease present in a location at any time point

• 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

• 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.

• 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%

• 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)

• 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)

• Definition
  
  • magnitude of the difference in the means of the control and experimental groups in a study with respect to the pooled standard deviation

• Definition
  
  • an estimate of the variability of each individual data point from the mean

• 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

• 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 

• Definition
  • the interval that will include a specific parameter of interest, if the experiment is repeated 
  • usually set at 95% by convention
    

• 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

• 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.

• 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