mean: true average median: middle number once ranked mode: most repetitive range : difference...

Statistics for AKT

Mean: true average Median: middle number once ranked Mode: most repetitive Range : difference between largest and smallest.

Discriptive Statistics

Find out the Mean, Median, Mode and Range for following.  8, 9, 9, 10, 11, 11, 11, 11, 12, 13

The mean is the usual average: (8 + 9 + 9 + 10 + 11 + 11 + 11 + 11 + 12 + 13) ÷ 10 = 105 ÷ 10 = 10.5

The median is the middle value. In a list of ten values, that will be the (10 + 1) ÷ 2 = 5.5th value which will be 11.

The mode is the number repeated most often. 11

The largest value is 13 and the smallest is 8, so the range is 13 – 8 = 5.

Normal Distribution: Mean=Median=Mode Positive Skewed: Mean>Median>Mode Negative Skewed: Mean<Median<Mode

Skewed Data

Test Result Disease Present Disease Absent

Positive TP FP

Negative FN TN

Sensitivity: How good is the test at detecting those with the condition

TRUE POSITIVESACTUAL NUMBER OF CASES

Specificity: How good is the test at excluding those without the condition

TRUE NEGATIVESACTUAL NUMBER OF PEOPLE WITHOUT CONDITION

Sensitivity and Specificity

Positive Predictive Value: How likely is a person who tests +ve to actually have the condition

TRUE POSITIVESNUMBER OF PEOPLE TESTING POSITIVE

Negative Predictive Value: How likely is a person who tests –ve to not have the condition

TRUE NEGATIVESNUMBER OF PEOPLE TESTING NEGATIVE

Predictive Values

Incorporates both sensitivity and specificity

Quantifies the increased odds of having the disease if you get a positive test result, or not having the disease if you get a negative test result.

Positive Likelihood ratio:Sensitivity

(1 – Specificity)

Negative Likelihood ratio:(1-Sensitivity)

Specificity

Likelihood Ratios

Odds are a ratio of the number of people who incur a

particular outcome to the number of people who do not incur the outcome.

NUMBER OF EVENTS NUMBER OF NON-EVENTS

Odds: what are the chances

Odds ratio:

The odds ratio may be defined as the ratio of the odds of a particular outcome with experimental treatment and that of control.

Odds ratios are the usual reported measure in case-controlstudies.

It approximates to relative risk if the outcome of interest israre.

ODDS IN TREATMENT GROUPODDS IN CONTROL GROUP

For example, if we look at a trial comparing the use of paracetamol for dysmenorrhoea compared to placebo we may get the following results

Total no of Patients

Pain relief achieved

Paracetamol 60 40

Placebo 90 30

The odds of achieving significant pain relief with paracetamol = 40 / 20 = 2

The odds of achieving significant pain relief with placebo = 30 / 60 = 0.5

Therefore the odds ratio = 2 / 0.5 = 4

Prevalence: rate of a disorder in a specified population

Incidence: Number of new cases of a disorder developing over a given time (normally 1 year)

Incidence and Prevalence

Questions

Relative risk (RR) is the ratio of risk in the experimental group (experimental event rate, EER) to risk in the control group (control event rate, CER).

Relative risk is a measure of how much a particular risk factor (say cigarette smoking) influences the risk of a specified outcome such as lung cancer, relative to the risk in the population as a whole.

Absolute risk: Risk of developing a condition

Relative risk: Risk of developing a condition as compared to another groupEVENTS IN CONTROL GROUP – EVENTS IN TREATMENT

GROUP EVENTS IN CONTROL GROUP

X 100- My lifetime risk of dying in a car accident is 5% - If I always wear a seatbelt, my risk is 2.5%

- The absolute risk reduction is 2.5%- The relative risk reduction is 50%

For example, if we look at a trial comparing the use of paracetamol for dysmenorrhoea compared to placebo we may get the following results

Total no of Patients

Pain relief achieved

Paracetamol 100 60

Placebo 80 20

Experimental event rate, EER = 60 / 100 = 0.6

Control event rate, CER = 20 / 80 = 0.25

Therefore the relative risk = EER / CER = 0.6 / 0.25 = 2.4

Relative risk reduction (RRR) or relative risk increase (RRI) is calculated by dividing the absolute risk change by the control event rate Using the above data,

RRI = (EER - CER) / CER (0.6 - 0.25) / 0.25 = 1.4 = 140%

Relative Risk Reduction

Numbers needed to treat (NNT) is a measure that indicates how many patients would require an intervention to reduce the expected number of outcomes by one

It is calculated by 1/(Absolute risk reduction)

Absolute risk reduction = (Experimental event rate) - (Control event rate)

Numbers needed to treat and absolute risk reduction

A study looks at the benefits of adding a new anti platelet drug to aspirin following a myocardial infarction. The following results are obtained:

Percentage of patients having further MI within 3 months Aspirin 4% Aspirin + new drug 3%

What is the number needed to treat to prevent one patient having a further myocardial infarction within 3 months?

NNT = 1 / (control event rate - experimental event rate) 1 / (0.04-0.03) 1 / (0.01) = 100

Remember that risk and odds are different.

If 20 patients die out of every 100 who have a myocardial infarction then the risk of dying is 20 / 100 = 0.2 whereas the odds are 20 / 80 = 0.25.

The null hypothesis is that there are no differences between two groups.

The alternative hypothesis is that there is a difference.

Null Hypothesis

Type 1 error:- Wrongly rejecting the null hypothesis- False +ve

Type II error:- Wrongly accepting the null hypothesis- False -ve

Study Error

Probability of establishing the expected difference between the treatments as being statistically significant- Power = 1 – Type II error (rate of false –ve’s)

Adequate power usually set at 0.8 / 80%

Is increased with- increased sample size- increased difference between treatments

Power of the study

A result is called statistically significant if it is unlikely to have occurred by chance

P values

- Usually taken as <0.05

- Study finding has a 95% chance of being true

- Probability of result happening by chance is 5%

Significance

Statistical Tests

1. Parametric / Non-parametricParametric if: - Normal distribution

- Data can be measured

2. Paired / Un-pairedPaired if data from a single subject group (eg before and after intervention)

3. Binomial – ie only 2 possible outcomes

Types of Tests

Student’s T-test- compares means- paired / unpaired

Analysis of variance (ANOVA)- use to compare more than 2 groups

Pearsons correlation coefficient- Linear correlation between 2 variables

Parametric Tests

Mann Whitney- unpaired data

Kruskal-Wallis analysis of ranks / Median test

Wilcoxon matched pairs- paired data

Friedman's two-way analysis of variance / Cochran Q

Spearman or Kendall correlation- linear correlation between 2 variables

Non Parametric Data

Compares proportions

Chi squared ± Yates correlation (2x2)

Fisher’s exact test- for larger samples

Binominal Data (non-parametric)

The standard deviation (SD) represents the average difference each observation in a sample lies from the sample mean

SD = square root (variance)

Standard Deviation

In statistics the 68-95-99.7 rule, or three-sigma rule, or empirical rule, states that for a normal distribution nearly all values lie within 3 standard deviations of the mean

About 68.27% of the values lie within 1 standard deviation of the mean.

Similarly, about 95.45% of the values lie within 2 standard deviations of the mean.

Nearly all (99.73%) of the values lie within 3 standard deviations of the mean.

Normal Distribution

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