Multivariate W1-3

Measure the strength of patterns in our data

One of two main goals of multivariate analysis.

Determine if this pattern is strong enough to be believed

The second of two main goals of multivariate analysis.

Data

Information collected to gain knowledge about a field or to answer a question of interest.

Design (in statistics)

Choosing subjects.

Description (in statistics)

Summarizing data.

Inference (in statistics)

Making predictions about a population based on a sample.

Population (statistical definition)

Total set of subjects of interest.

Sample (statistical definition)

Subset of the population on which a study collects its data.

Parameter

Numerical summary of a population. Example: percentage of all adult Americans.

Statistic

Numerical summary of a sample. Example: percentage of adult Americans in a specific location.

Nominal level of measurement

Data that consist of names, labels, or categories only; qualitative and cannot be ranked or ordered.

Ordinal level of measurement

Qualitative data that can be arranged in some order (low to high); no quantitatively fixed space between items.

Interval level of measurement

Quantitative data where intervals are meaningful but ratios are not; has an arbitrary zero point.

Mean

Average value; sum of all values divided by total number of values.

Advantage of mean

Easy to interpret; foundational statistic usable in many procedures; sample mean is best estimate of population mean.

Disadvantage of mean

Requires interval or ratio level data; can be highly influenced by outliers, especially in small sample sizes.

Median

Exact middle value in a sorted data set.

Data requirement for median

Requires ordinal data.

Advantage of median

True model of central tendency because it is always in the middle; not strongly influenced by outliers.

Mode

The most common value in a data set.

Standard deviation

A measure of variability for interval variables equal to the square root of the variance.

Best summary for nominal data

Mode.

Best summary for ordinal data

Mode, Median, and Range.

Best summary for interval-ratio data

Mean, Median, Mode, Range, and Standard Deviation.

Normal distribution characteristics

Symmetrical/even distribution.

Central limit theorem

Things that start as not normal (bi-modal) can become normal with enough added data; sample looks like population as you collect more data.

Z-score

A way to identify one's position and location in a distribution of data; changes a raw score to a standardized or normalized score.

Z-score formula

The individuals data (Xi) - the sample mean (x̅), divided by the samples data’s standard deviation (sx)

Purpose of z-scores

Crucial to make comparisons across variables with different units (income, education years, number of children).

Inferential statistics definition

We rarely observe population parameters, only sample statistics; to draw inferences, samples must be representative.

Sampling error

Almost every sample will miss the true population parameter by a little.

Sampling distribution

Theoretical distribution of a statistic for all possible samples of a given size (N); does not exist empirically.

Sampling with replacement

Choosing a subject randomly as first member, then putting them back before choosing the next member.

Mean of sampling distribution

Equal to the true population mean (

Standard error

Standard deviation of the sampling distribution; equals population standard deviation divided by square root of N (

Population distribution

A real distribution representing characteristics of all members of the population of interest.

Sampling distribution (summary)

A theoretical probability distribution representing results of all possible samples drawn from the population.

Sample distribution

A real distribution describing characteristics of a sample.

Central Limit Theorem summary

Most random samples with relatively large N are close to true population mean; larger N means smaller standard error and closer clustering to true parameter.

Point estimate

Sample statistic used to estimate the exact value of a population parameter (mean, proportion, etc.).

Confidence interval

Range built around a sample statistic within which the population parameter is likely to fall.

Confidence interval width principle

Confidence in a range goes up the wider it is because it can account for more possible values.

Default confidence interval used in class

95 percent confidence interval.

When to reject null hypothesis

p<0.05, it is unlikely the observed difference is by random chance.

Two sample t-test

Compare differences between two sample statistics from two mutually exclusive, independently and randomly selected groups.

Example of two sample t-test

Gender wage gap between men and women.

ANOVA (Analysis of Variance)

Compares differences across more than two groups; essentially a 3+ group t-test.

ANOVA logic

Uses a sampling distribution of variation in means; decomposes variance to compare between-group differences to within-group differences.

ANOVA example age and capital punishment

Different age groups should have different support levels; a particular age group should have similar support levels.

Three steps of quantitative research

1. Have research idea and hypothesize questions; 2. Find appropriate data/variables; 3. Choose statistical analysis and interpret findings.

Three bivariate associations

X1→Y (direct cause); X1→X2 (cause to mediator); X2→Y (mediator to outcome).

Mediating variable example

Does racism cause educational barriers which then causes Y.