statistics

Question 1

What is a variable?

Answer 1

Measure of any single characteristic

Can be assigned a number or category

Can be discrete or continuous

Question 2

What is a discrete variable ?

Answer 2

A numeric variable for which we can list the possible values

Can be qualitative or descriptive

Can be dichotomous or polychotomous

Can be nominal or ordinal

Question 3

What are continuous variables ?

Answer 3

Can be quantitative or numerical

Truly measurable

Question 4

What are parametric statistical tests?

Answer 4

Used for continuous variables

makes assumptions about frequency distribution of data

more powerful than non parametric tests

Question 5

What are non-parametric tests?

Answer 5

used for discrete variables

makes no assumption about the frequency distribution of data

Question 6

What is normal distribution and describe how it looks on a graph?

Answer 6

special density curve with a bell shape

Data clusters around a central value

Continuous probability distribution

Normal distribution is closer to a bell shaped curve

Data cluster around a central value

curve is symmetric

curve not too peaked or too flat

Question 7

What are the parameter for normal distribution of a population?

Answer 7

Central tendency:
Average or arithmetic meaning - μp= ∑x/n

Dispersion:
Standard deviation -
σp= √∑(μp-x)^2/n

Question 8

What are the parameters of normal distribution of a sample?

Answer 8

Central tendency:
- Average or arithmetic meaning:
μs = ∑(μs - x)^2/(n-1)

Dispersion
- Standard deviation:
σs= √∑(μs-x)^2/(n-1)

Question 9

How is standard deviation derived?

Answer 9

1. Sum each observation subtracted from the mean: ∑(μ-x)
2. Subtract sums of squares:
   SS = ∑(μ-x)^2
3. Mean square:
   MS = ∑((μ-x)^2)/n
4. Standard deviation:
   σ= √∑((μ-x)^2)/n

Question 10

How are the mean and standard deviation shown in publications?

Answer 10

Use mean, standard deviation (σs, μs)

Mean +- standard deviation (σs +- μs)

+- should be confined to confidence intervals

Question 11

What is the coefficient of variation (CV) and its equation?

Answer 11

Standard deviation expressed as a percentage of mean is CV

CV = σ/μ * 100
σ = standard deviation
μ = mean(population sample)

Question 12

What is the equation of normal distribution?

Answer 12

Normal distribution is defined completely by its mean and standard deviation

f(x) = (1/σ√2pi) e^(-(x-μ^2/2σ^2))

F(x) = frequency of a particular value of x

σ = Standard deviation

e = exponent

μ = mean

Question 13

how do you convert from a member of the original distribution to a member of the standard normal distributions? (formula)

Answer 13

z = +-(x - μp)/σp

z = A value of the standard normal distibution

x = your original observation

μp =Established population mean

σp = Established standard population deviation

Measured x is atypical if z>_ 1.96

Question 14

Name some properties of normal distribution

Answer 14

Most visual functions follow a normal (Gaussian) distribution.

The density of observed values is greatest near the centre of the
distribution.

Outliers, at the edge (or tails) of the distribution, are relatively rare.

The normal distribution allows mathematical prediction of the chance
of a particular data value occurring.

Parametric statistical tests assume normal distribution.

Question 15

What does a red bell shaped distribution curve depict ?

Answer 15

Normal distribution, F(x), of individual values

its dispersion is represented by sigma

Question 16

What does a blue bell shaped curve depict?

Answer 16

Normal distribution of sample means, F(x*) or F(mean)

Its dispersion is represented by 𝝈/√n

where n is the size of the sample.

Question 17

What is the central limit theorem?

Answer 17

sample means from a normal distribution of individual values
are themselves normally distributed.

means of non-normal distributions will also be normally
distributed as long as the samples are large enough

Question 18

What is the 4 step process to calculate the confidence limits for a sample mean?

Answer 18

1. There is a 95% chance that an individual observation,
   belonging to a normal distribution, lies between:
   µp ± 1.96𝝈p ………. (1)
2. There is a 95% chance that the sample mean, belonging to
   data showing a normal distribution, lies between:
   µp ± 1.96𝝈p/√n ………. (2)
3. Rearrangement of equation (2) shows that there is a 95% chance
   that the population mean lies between:
   µs ± 1.96𝝈p /√n ………. (3)
4. The 95% confidence limits of a sample mean equals:
   µs ± t (𝝈s /√n) ………. (4)

Question 19

What standard deviation ?

Answer 19

the variability of a single set of individual measurements.

Question 20

Whats standard error of the mean?

Answer 20

the degree of error associated with estimating a sample mean.

Question 21

What is the confidence liits?

Answer 21

the limits within which a population mean would fall.

Question 22

When a bell curve is not bell shaped what terms are used instead?

Answer 22

Positive skew

Negative skew

Question 23

Whats a positive skew?

Answer 23

tail is pulled out towards the upper end of the distribution

curve peak to the left

Higher mean than median

Question 24

Whats a negative skew?

Answer 24

tail is pulled toward the lower end of the distribution

Curve/peak is on the left side

lower mean than median

Question 25

what is kurtosis?

Answer 25

the degree to which a curve is peak or flat

the greater the kurtosis the larger the peak is

Question 26

What is negative kurtosis?

Answer 26

Compared to normal distribution

• Flat-topped
• Less observations near the mean

Question 27

Whats is positive Kurtosis?

Answer 27

Compared to normal distribution

• Peaked
• More observations near the mean

Question 28

What are the two other measures of central tendencies?

Answer 28

Mode

Median

Question 29

What is mode?

Answer 29

Value of variable 𝒙 with the highest frequency.

The maximum point of the curve.

The most common value.

Can be more than one mode.

Question 30

What is the Median?

Answer 30

The middle value of 𝒙 if all values listed in order.

Used for non-parametric tests.

Question 31

Give alternative measures of dispersion?

Answer 31

Percentiles

Interquartile range

Question 32

What is a percentile?

Answer 32

% of observations below a particular value when ranked in descending order.

50% percentile represents the median.

Question 33

What is a range?

Answer 33

An alternate measure of dispersion.

Difference between the largest score and the smallest score
of a distribution

Indicates where any value may lie, including outliers.

Question 34

What is interquartile range ?

Answer 34

The numerical distance between the 25th and 75th percentiles.

Includes 50% of distribution cantered on the median value.

Applicable to data that is not normally distributed.

Question 35

Explain how the assessment of risk is carried out for a cohort study

Answer 35

Risk factor present - diseases present - a
disease absent - b
total - a+b

Risk factor absent
disease present - c
disease absent - d
total - c+d

total
- a+c
- b+d
- a+b+c+d

Incidence = new occurrences of disease in population at risk during study.

• Incidence in exposed group = a / (a + b)
• Incidence in unexposed group = c / (c + d)

Relative risk (RR) = ratio of incidence in exposed group to incidence in unexposed group
- RR = [a / (a + b)]/[c / (c + d)]

95% confidence limits
- Variance = √[1/a + 1/b + 1/c + 1/d]
- Lower limit = RR∙e (-1.96∙variance)
- Upper limit = RR∙e (1.96∙variance)

Question 36

Show the equations used for case control studies

Answer 36

Risk factor present - Case(disease) - a
Control(no disease) - b
total - a+b

Risk factor absent
Case - c
Control - d
total - c+d

total
- a+c
- b+d
- a+b+c+d

Prevalence = proportion of population with disease at one point in.

Odds ratio (OR) = odds of exposure in cases divided by odds of exposure in controls
- OR = (a / c)/(b / d) = (ad) / (bc)

95% confidence limits
- Variance = √[1/a + 1/b + 1/c + 1/d]
- Lower limit = OR∙e (-1.96∙variance)
- Upper limit = OR∙e (1.96∙variance)

Question 37

what is validity, related terms, and how is it assesses?

Answer 37

Related terms: comparability, accuracy

Ability to correctly measure what is supposed to be measured

Assessed using: Bland-Altman, limits of agreement, kappa

Question 38

what is Discriminative ability, related terms, and how is it assesses?

Answer 38

Ability to distinguish normal from abnormal

Assessed using: sensitivity, specificity, likelihood ratios, predictive values and ROC (Receiver Operating
characteristic) curves

Question 39

what is repeatability, related terms,its conditions ,and how is it assesses?

Answer 39

Related terms: precision, reliability, variability, reproducibility

Consistency of repeated measurements
- Repeatability conditions: measurements repeated with same method on same patients in same clinic
by same clinician using same equipment

• Reproducibility conditions: measurements repeated with same method on same patients but in
  different clinics by different clinicians using different equipment

Assessed using: Bland-Altman plots, coefficient of repeatability, kappa

Question 40

Show the evaluation of false-positive and false negative errors

Answer 40

Test result positive
Disease present - a
Disease absent - b
total - a + b

Test result negative
Disease present - c
disease absent - d
total - c+d

Total
disease present - a+c
disease absent - b+d
total - a+b+c+d

b = false positive test result

c = false negative test result

a + c = all people with disease

b + d = all people without disease

False-positive error rate (type I or alpha)

• data indicate that disease is present when it is absent
• b / (b + d)

False-negative error rate (type II or beta)

• data indicate that disease is absent when it is present
• c / (a + c)

Question 41

What is sensitivity?

Answer 41

Ability to detect disease when it is present

a / (a + c)

a = true positive test result
d = true negative test result
a + c = all people with disease
b + d = all people without disease

Depends on minimum false-negative errors

Sensitivity + false negative error rate = 1

‘SnNOut’ – sensitive test when Negative rules Out disease

Question 42

What is specificity?

Answer 42

Ability to indicate no disease when none is present.

d / (b + d)

a = true positive test result d = true negative test result
a + c = all people with disease
b + d = all people without disease

Depends on minimum false-positive errors

Specificity + false positive error rate = 1

‘SpPln’ – specificity test when Positive rules In disease

Question 43

Explain the evaluation test for bayes theorem, predictive values, pre- and post- test propapility

Answer 43

Test result positive
Disease present - a
Disease absent - b
total - a + b

Test result negative
Disease present - c
disease absent - d
total - c+d

Total
disease present - a+c
disease absent - b+d
total - a+b+c+d

a = true positive test result
d = true negative test result
a + c = all people with disease
a + b = all people with a positive test result
c + d = all people with a negative test result
a + b + c + d = all people tested

Pre-test or prior probability (prevalence)
- (a + c) / (a + b + c + d)

Positive predictive value
- Probability of having disease if the test result is positive
- a / (a + b)
- Is Bayes’ theorem rewritten
- Is post-test or posterior probability if the test result is positive

Negative predictive value
Probability of not having disease if the test result is negative
- d / (c + d)
- 1 – negative predictive value is post-test or posterior probability if the test result is negative: effectively c / (c + d)

Predictive values
- Depend on prevalence

Question 44

Explain the evaluation of tests for likelihood ratios, ore- and post-test odds

Answer 44

Test result positive
Disease present - a
Disease absent - b
total - a + b

Test result negative
Disease present - c
disease absent - d
total - c+d

Total
disease present - a+c
disease absent - b+d
total - a+b+c+d

a = true positive test result
d = true negative test result
a + c = all people with disease
a + b = all people with a positive test result
c + d = all people with a negative test result
a + b + c + d = all people tested

Positive likelihood ratio
- Ratio of sensitivity to false-positive error rate
- [a / (a + c)] / [b / (b + d)] = (a / b) / [(a + c) / (b + d)]
- The larger the value, the better the test

Negative likelihood ratio
- Ratio of false-negative error rate to specificity
- [c / (a + c)] / [d / (b + d)] = (c / d) / [(a + c) / (b + d)]
- The lower the value, the better the test

Likelihood values
- Independent of prevalence

Pre-test odds
- Odds of disease before the test
- (a + c) / (b + d)
- Also equals pretest probability / (1-pretest probability)

Post-test odds
- Odds of having disease after having been tested positive
- Pretest odds multiplied by positive likelihood ratio
- [(a + c) / (b + d)].{ (a / b) / [(a + c) / (b + d)]} = a / b
- Post-test probability = post-test odds / (post-test odds + 1)

Question 45

what are ROC curves and what does it show?

Answer 45

Plot of sensitivity versus specificity.

Analysis is considered as a reliable method
for evaluating the diagnostic ability.

It shows the trade-off between sensitivity
and specificity

Question 46

Explain the Kappa evaluation

Answer 46

Used to test against a gold standard

Table
Test 1
Test 2
- test positive - test positive = a
- test positive - test negative = b
total = a+b

test negative - test positive = c
test negative - test negativ = d
Total = c+d

Total
positiv - a+c
negative- b+d
total = a+b+c+d

Observed agreement (Ao) = a + d

Maximum possible agreement (N) = a + b + c + d

Overall Percentage agreement = 100 [(a + d) / (a + b + c + d)]

Cell a agreement expected by chance = [(a + b)(a + c)]/(a + b + c + d)

Cell d agreement expected by chance = [(c + d)(b + d)]/(a + b + c + d)

Total agreement by chance (Ac)

Cell a agreement expected by chance + cell d agreement expected by chance

Intraclass Kappa = (Ao - Ac)/(N - Ac)