Lecture 5 - Foundations of quantitative research - probability, normal distribution and central limit theorem

Question 1

Reliability and Validity

Answer 1

• Quantitative research is about measurement
– Reliability
• Consistency and repeatability of a measure
• Types of reliability
– Intra-rater, inter-rater and test-retest
– Validity
• Accuracy
• Does an instrument measure what it intends to measure
• Types
– Face, content and criterion validity

Question 2

Internal and External Validity

Answer 2

• Internal validity
– Relates to the conduct of the research
– Can you be confident that the research was well conducted so that the effect you notice is from the exposure or intervention (causal relationship)
• External validity
– Generalisability of the research findings from the sample to the reference population

Question 3

What can affect internal validity?

Answer 3

• Chance (imprecision) – Random error
– Cannot be eliminated but minimised
– Usually through having sufficient sample size
• Bias
– Systematic error
– Errors in the way research was undertaken – Can be eliminated
• Confounders
– Variables that were not taken into account and
hence influence outcome

Question 4

Minimising the role of chance

Answer 4

Minimising the role of chance
• Using proper sampling strategy
– Avoiding convenience sampling
– Random (from the telephone book) or systematic sampling (every 5th person coming through the door)
• Having adequate sample size
– We can work out the number we need in our research
• Errors in sampling
– Type 1 – finding a significant result/difference when it is not there
– Type 2 – finding no significant result/difference when it is there

Question 5

What is probability?

Answer 5

• The number of times an outcome occurs in the total number of trials.
– If A is the outcome, the probability of A is denoted by P(A)
• Measure of the likelihood that an event in the future will happen

Question 6

How to interpret probability

Answer 6

• If P(A) equals zero, event A will almost definitely not occur.
• If P(A) is close to zero, there is only a small chance that event A will occur.
• If P(A) equals 0.5, there is a 50-50 chance that event A will occur.
• If P(A) is close to one, there is a strong chance that event A will occur.
• If P(A) equals one, event A will almost definitely occur.

Question 7

Basic definitions in probability

Answer 7

• Experiment
– Planned process of data collection
• Outcome
– The result of an experiment or trial
• Events
– A single outcome or a set of outcomes from an experiment

Question 8

Basic Definitions in probability: Types of events

Answer 8

• Complementary events (A)
– Probability that an event will not happen
– An event opposite to the event of interest
– Example: Coin tossing (A is the even head is the outcome)
• A is the event that tail is the outcome.
• Mutually exclusive events
– Two or more events are mutually exclusive if the occurrence of one precludes the occurrence of the others
– Two events cannot occur at the same time
• Independent events
– Two different events are independent if the outcome of one event has no effect on the outcome of the second.

Question 9

Types of probability

Answer 9

• Marginal Probability
– Probability of the occurrence of a single event – p(A)
• Joint Probability
– The probability that two events will occur simultaneously – p(A and B)
• Conditional Probability
– The probability of one event given that another events has occurred.
– P(A I B)

Question 10

Diagnostic Accuracy

Answer 10

• Diagnostic testing is an important part of health care
• There is not a single test that is always 100% accurate
– Except possibly one
• So, as part of diagnostic testing, we have to work out how confident we are in believing the results of a diagnostic test

Question 11

Diagnostic Accuracy: Sensitivity

Answer 11

• Sensitivity (Sn)
– The proportion of people with disease who will have a
positive result
– If the test is highly sensitive and the test result is positive you can be nearly certain that they have disease
• Those who have a negative finding are likely not to have the disease
– TP = (TP/(TP+FN)*100
– Snout - A Sensitive test helps rule out disease

Question 12

Diagnostic Accuracy: Specificity

Answer 12

• Specificity
– The probability that the diagnostic test is negative in patients who do not have the disease
– If the test result for a highly specific test is negative, then you can be nearly certain that they don’t have the disease
• Those who test positive are likely to have the disease – TN = TN/(TN+FP)*100
– Spin - A very Specific test rules in disease

Question 13

Normal Distribution

Answer 13

• A symmetric, bell shaped probability distribution with mean μ and standard deviation δ.
• If observations follow a normal distribution, the interval (μ±2δ) contains 95% of the observation.
Characteristics:
-unimodal
-symmetric
-asymptotic

Question 14

Characteristics of normal distribution

Answer 14

• Symmetrical or two halves identical
• Total area under the curve is 1
• Mean, median and mode are equal
• Theoretically, the curve extends to ±∞

Question 15

Standard Normal Probability Distribution (z Distribution)

Answer 15

• Normal probability curve that has a mean of 0 and standard deviation of 1.
Z = how far is the data from the population mean (by standard deviations)

Question 16

Central Limit Theorem

Answer 16

• It is the cornerstone of statistics and research
• Useful for any type of data analysis
• Address a simple question
– How the we know use the data from the sample and make inference for the reference population?
• A theorem that states that the distribution of the means is approximately normal if the sample size is large enough, regardless of the underlying distribution of the original measurements.
• Predicts that regardless of the distribution of the parent population:
– The mean of the population of means is always equal to the mean of the parent population from which the samples were drawn.
– The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size
– The distribution of means will increasingly approximate a normal distribution as the sample size increases.