biostat module 2: unit 3 (module)

PROBABILITY

Appropriately conducting and interpreting biostatistical applications require attention to a number of important issues.
● Clearly defining the objective or research question;
● Choosing an appropriate study design;
● Selecting a representative sample, and ensuring that the sample is of sufficient size;
● Carefully collecting and analyzing the data;
● Producing appropriate summary measures or statistics;
● Generating appropriate measures of effect or association;
● Quantifying uncertainty;
● Appropriately accounting for relationships among characteristics; and,
● Limiting inferences to the appropriate population.

Probability theory

The collection and summarization of data and the drawing of inferences require knowledge of biostatistics principles grounded in _____.

Probability sampling

each member of the population has a known probability of being selected.

Non-probability sampling

each member of the population is selected without the use of probability.

Probability sampling

includes simple random sampling, systematic sampling, and stratified sampling.

Non-probability sampling

includes convenience sampling and quota sampling.

Probability

refers to the proportion of times an event is expected to occur in the long run. It is a number that reflects the likelihood that a particular event, such as sampling a particular individual from a population into a sample, will occur. It is a mathematical tool that helps us understand and address chance. It permits dealing with random variability in constructive ways.

Random variable

is a numerical quantity that takes on different values depending on chance. This is not totally different to the initial definition of variable but, if the process that produced the value of that variable is random.

Discrete random variables

form a countable set of possible values

Continuous random variables

form an unbroken continuum of possible values.

Event

is an outcome or set of outcomes for a random variable.

Binomial probability distributions

are the most common type of probability distribution that applies to discrete random variables in which:
● There are n independent observations.
● Each observation can be characterized as a "success" or "failure".
● The probability of "success" for each observation is a constant p.

Binomial random variables

have two parameters: n (number of observations) and p (probability of success for each observation). The cumulative probability of an event is the probability of observing given value x or less. Again, we focus on events where the outcome could be binomial or dichotomous and as denoted by "success" or "failure" as mentioned above.

Normal (Gaussian) distributions

are the most common type of probability distribution that applies to continuous random variables. They are recognized by their symmetry, bell shape, points of inflection at population mean minus one standard deviation and population mean plus one standard deviation, and horizontal asymptotes as they approach the X axis on either side.

Z-distribution

Many random variables encountered in nature are not Normal. However, these random variables can be made Normal by re-expressing them with mathematical transformation such as logs, exponents, powers, roots, and quadratics. To determine Normal probabilities, values are first standardized to transform it to a Normal scale with mean 0 and standard deviation 1.

Z-score

tells the distance the value falls from the mean in standard deviation units.

Positive z-scores

Values that are larger than the mean.

Negative z-scores

Values that are smaller than the mean.

Statistical inference

the act of using data in a particular sample to make generalizations about the population from which it came from.

Standard error or the mean

The standard deviation of the sampling distribution of sample mean. It is an estimate of the sample mean's precision as an estimate of the population mean. Inversely proportional to the square root of the sample size.

Square root law

Because of it, the standard error of the mean gets smaller and smaller as the sample size gets larger and larger. Therefore, sample means based on large n are more likely to fall close to the true value of the population mean than means on small n (all other things being equal).

Central limit theorem

When sample size n is large, the sampling distribution of the sample mean tends toward Normality even when the population is not Normal.

Hypothesis testing

uses a deductive procedure to judge claims about parameters. A specific hypothesis is made about a population parameter and a sample statistic is used to determine the chance whether the hypothesis is true.

Small P-value

indicates that observed data are unlikely to have come from the distribution suggested by the null hypothesis.

Alpha

sets the standard for how extreme the data must be before we can reject the null hypothesis and is predetermined when designing studies.

P-value

computed from data and would indicate how extreme the data are.

Type I error

an erroneous rejection of a true null hypothesis.

Type II error

an erroneous retention of a false null hypothesis.

Estimation

We use sample statistics to produce estimates about unknown population parameters.

Point estimation

provides a single estimate of the parameter

interval estimation

provides a range of values (confidence intervals) that seek to capture the parameter.

Confidence level of a confidence interval

refers to the success rate of the method in capturing the parameter it seeks.

Systematic errors

come in these three general forms: confounding, information bias, and selection bias.

Confounding

is especially problematic in non-experimental studies. It derives from the mixing together of the effects of the explanatory variable and extraneous variables lurking in the background.

Confounders

Extraneous variables that cause confounding.

Information bias

arises from defects in measurement. With categorical data, this corresponds to misclassification of the explanatory variable or response variable or both. Such misclassifications may be either differential or non-differential.

Differential misclassifications

affect some groups more than others. These biases can lead to false positive or false negative results

Non-differential misclassification

occurs to the same extent in the groups being compared. These biases may result either toward acceptance of the null hypothesis or not at all.

Selection bias

due to the manner in which subjects are selected for the study. The lack of eligibility criteria and/or the deliberate assignment of study participants into groups to bring about expected results are some examples of selection bias sources.