Updated: 5/22/2020

# Statistic Definitions

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Introduction
• This topic covers the following statistical principles
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 positivea (6) false positiveb test neg false negativec  (4) true negatived 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 positivea false positiveb (5) test neg false negativec true negatived (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 positivea false positiveb test neg false negativec true negatived

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 positivea false positiveb test neg false negativec true negatived
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 positivea (9) false positiveb (4.5) test neg false negativec (1) true negatived (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 positivea (80) false positiveb (5) test neg false negativec (20) true negatived (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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Questions (42)
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(OBQ13.116) A prospective randomized trial is conducted to test the efficacy of Vitamin C versus placebo in treating patients who develop chronic regional pain syndrome (CRPS) after distal radius fractures. At first follow-up, the rates of CRPS are 1% and 9% in the study and placebo group, respectively. Which statistical test is most appropriate to determine significance? Tested Concept

QID: 4751
1

Single factor analysis of variance

6%

(245/4166)

2

Chi-square test

51%

(2121/4166)

3

Student t-test

40%

(1651/4166)

4

Mann-Whitney rank sum test

2%

(67/4166)

5

Wilcoxon rank sum test

1%

(26/4166)

L 4 A

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(OBQ13.126) A prospective, randomized controlled trial of 150 patients undergoing total hip arthroplasty is performed to test whether repair of the capsule during a posterior approach reduces post-operative dislocations in the first three months. The study found no difference in dislocation rate if the capsule was repaired versus not repaired (p = .34). Subsequently, a multicenter follow-up study of 2000 patients showed that repairing the capsule led to a decreased dislocation rate in the first three months (p = .03). Assuming the second study reflects reality, which of the following errors occurred in the first study? Tested Concept

QID: 4761
1

Observer bias

2%

(61/2948)

2

Type-II error

66%

(1960/2948)

3

Alpha error

7%

(200/2948)

4

Type-I error

21%

(605/2948)

5

Confounding error

3%

(92/2948)

L 2 A

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(OBQ12.200) The ability of a study to detect the difference between two interventions if one in fact exists describes which of the following? Tested Concept

QID: 4560
1

Positive predictive value

12%

(507/4291)

2

Hawthorne effect

2%

(88/4291)

3

Effect size

2%

(99/4291)

4

Power

67%

(2854/4291)

5

P value

17%

(709/4291)

L 2 A

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(OBQ12.146) While conducting a retrospective review of patients undergoing two different techniques for open reduction and internal fixation of ankle fractures, the investigator would like to assess whether there is any significant difference between the mean patient age in the two groups. The two groups are normally distributed. Which of the following tests would be most appropriate? Tested Concept

QID: 4506
1

Student t-test

66%

(1892/2852)

2

Analysis of Variance (ANOVA)

13%

(359/2852)

3

Fisher exact test

3%

(99/2852)

4

Kruskal-Wallis test

1%

(16/2852)

5

Chi-square test

16%

(462/2852)

L 3 A

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(OBQ11.219) Which of the following best describes a Bonferroni correction? Tested Concept

QID: 3642
1

An analysis that starts with a particular probability of an event (the prior probability) and incorporates new information to generate a revised probability (a posterior probability)

18%

(459/2547)

2

Human behavior that is changed when participants are aware that their behavior is being observed.

10%

(247/2547)

3

Used to assess the relationship between two normally distributed continuous variables

5%

(116/2547)

4

A post-hoc statistical correction made to P values when several dependent or independent statistical tests are being performed simultaneously on a single data set

64%

(1634/2547)

5

The ability of a study to detect the difference between two interventions if one in fact exists

3%

(64/2547)

L 3 C

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(OBQ11.42) The sensitivity of a serologic assay is defined as which of the following? Tested Concept

QID: 3465
1

True positives / (true positives + true negatives)

9%

(138/1541)

2

False negatives / (false negatives + true positives)

2%

(28/1541)

3

False positives / (false positives + false negatives)

1%

(18/1541)

4

(True positives + false positives) / (true negatives + false negatives)

1%

(18/1541)

5

True positives / (true positives + false negatives)

86%

(1326/1541)

L 1 B

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(OBQ10.207) The chi-square test is considered the most appropriate statistical test to analyze categorical data, but is unreliable if there are less than 5 events in any of the groups or the sum of all cells is less than 50. Which test is preferred in place of the chi-square test when these small sample sizes are encountered? Tested Concept

QID: 3300
1

Fisher exact test

57%

(1564/2744)

2

Regression analysis

3%

(84/2744)

3

Two-sample t-test

15%

(424/2744)

4

Mann-Whitney test

11%

(292/2744)

5

Analysis of variance (ANOVA)

13%

(366/2744)

L 3 C

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(OBQ10.43) The statistical power of a study is best defined by? Tested Concept

QID: 3131
1

1 - probability of type-II (beta) error

72%

(1271/1764)

2

True positive/(true positive + false negative)

3%

(52/1764)

3

True negative/(false positive + true negative)

1%

(16/1764)

4

1 - probability of type-I (alpha) error

20%

(360/1764)

5

[True positive/(true positive + false negative)] / false-positive rate

3%

(55/1764)

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(OBQ10.153) A trial is peformed evaluating the use of ultrasound to diagnose meniscus tears in 100 athletes with knee pain. Figure A displays the data from the ultrasound examinations compared to the gold standard of arthroscopic diagnosis. The statistician calculates the following equation: 86/[86+4]= 95.5%. What statistical term does this equation best describe? Tested Concept

QID: 3241
FIGURES:
1

Sensitivity

11%

(377/3287)

2

Positive predictive value

7%

(214/3287)

3

Specificity

58%

(1907/3287)

4

Negative predictive value

23%

(748/3287)

5

Likelihood ratio

1%

(20/3287)

L 3 B

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(OBQ09.185) The positive predictive value is defined as which of the following? Tested Concept

QID: 2998
1

True positives / (true positives + true negatives)

11%

(106/956)

2

False negatives / (false negatives + true positives)

2%

(15/956)

3

False positives / (false positives + false negatives)

1%

(8/956)

4

(True positives + false positives) / (true negatives + false negatives)

3%

(27/956)

5

True positives / (true positives + false positives)

83%

(795/956)

L 1 B

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(OBQ09.125) What term in statistics defines accepting the null hypothesis when it is in fact not true? Tested Concept

QID: 2938
1

Type I error

31%

(682/2204)

2

Type II error

65%

(1425/2204)

3

Bias

2%

(42/2204)

4

Negative predictive value

2%

(38/2204)

5

Positive predictive value

0%

(8/2204)

L 3 B

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(OBQ08.218) Which of the following terms best describes the probability of finding a significant association in a research study when one truly exists? Tested Concept

QID: 604
1

Type-1 (alpha) error

9%

(62/668)

2

Type-2 (beta) error

6%

(42/668)

3

Power

67%

(446/668)

4

Alpha level

15%

(101/668)

5

Relative Risk

2%

(13/668)

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(OBQ08.71) The estimated range of values which likely includes the unknown parameter under investigation is defined as which of the following? Tested Concept

QID: 457
1

Standard deviation

10%

(73/758)

2

Mode

1%

(10/758)

3

Variance

6%

(48/758)

4

Confidence interval

82%

(622/758)

5

Incidence

0%

(2/758)

L 2 C

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(OBQ08.74) What is the equation for determining specificity of a clinical test? Tested Concept

QID: 460
1

True negatives divided by the sum of the true negatives and false positives

57%

(361/638)

2

True negatives divided by the sum of the true negatives and false negatives

15%

(95/638)

3

True positives divided by the sum of the true negatives and false positives

5%

(30/638)

4

True positives divided by the sum of the true positives and false negatives

13%

(85/638)

5

True positives divided by the sum of the true positives and false positives

10%

(61/638)

L 3 A

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(OBQ06.50) The definition of effect size is best described as which of the following? Tested Concept

QID: 161
1

Likelihood that a statistically significant difference would be found between 2 groups given that a difference truly did exist

12%

(154/1324)

2

Estimated magnitude of the difference in the means between two groups

61%

(806/1324)

3

Average of the squares of each value's deviation from the mean

3%

(38/1324)

4

Range within which it is probable that the true value lies for the whole population of patients

17%

(225/1324)

5

Probability of obtaining a result equal to or more extreme than what was actually observed assuming the null hypothesis is true

7%

(90/1324)

L 3 C

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(OBQ05.18) Which of the following defines the incidence of a disease? Tested Concept

QID: 55
1

The total number of cases of a disease in a city

2%

(10/632)

2

The number of new cases of a disease diagnosed during a specific time period

91%

(575/632)

3

The average number of cases of a disease per year over the last 10 years

1%

(9/632)

4

The number of existing cases of a disease divided by total population in a city

5%

(34/632)

5

The variability in the total number of disease cases between major US cities

0%

(1/632)

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(OBQ05.188) A prospective cohort study is performed looking at the relationship between blood transfusions and the risk of developing hepatitis C. In the transfused group (study group) of 595 patients, 75 patients develop hepatitis C. In the non-transfused group (control group) of 712 people, 16 people contract hepatitis C. What is the relative risk of developing hepatitis C with a transfusion. Tested Concept

QID: 1074
1

Incidence of study group (75/595) divided by incidence of control group (16/712)

78%

(920/1175)

2

Incidence of study group (16/595) divided by incidence of control group (75/712)

4%

(52/1175)

3

Prevalence of study group (75/595) divided by prevalence of control group (16/712)

14%

(162/1175)

4

Prevalence of study group (16/595) divided by prevalence of control group (75/712)

1%

(9/1175)

5

Total infected (75+16) divided by total population in study (595+712)

2%

(26/1175)

L 2 A

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(OBQ04.125) The paired Student's t-test is most appropriately used for which of the following? Tested Concept

QID: 1230
1

Determining if the medians are equal in two observed samples

7%

(45/618)

2

Discerning differences among a group of more than two means

9%

(58/618)

3

Illustrating an actuarial method of survival rates

1%

(9/618)

4

Refining a correlation coefficient among outlying observations

1%

(9/618)

5

Evaluating the difference between two observed means in matched groups

80%

(496/618)

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