Intro to Statistical Models

• This is a summary of statistics with Prof. Andy Field.

Aims

• What a statistical model is and why they are used should be understood.
• What the 'fit' of a model is and why it is important should be understood.
• Distinguish models for samples and populations.
• Identify the problems with NHST and modern approaches.
• Reporting confidence intervals and effect sizes should be understood.

The Research Process

• Data leads to an initial observation (research question), which leads to a theory.
• Variables are identified to generate hypotheses, leading to predictions.
• When measuring variables, data is collected to test predictions.
• The data is then analyzed by graphing data, and fit to a model.

Statistical Models

• Statistical models are used to build a representation of the real world.
• The "good fit" has the data points close to the predicted values.
• The “poor fit” has the data points far away from the predicted values.

Populations and Samples

• The population is the collection of units, such as people, plankton, etc. that findings or a statistical model are generalized to.
• A sample is a smaller collection of units from a population used to determine truths about that population.

The Equation

• The equation for statistics will be "Outcome = Model + error".

A Simple Statistical Model

• In statistics, models are fit to data to represent what is happening in the real world.
• The mean is a hypothetical value, it doesn't have to exist in the data set.
• As such, the mean is a simple statistical model.

The Mean

• The mean is the value from which the squared scores deviate least.

Calculating Error

• A deviation is the difference between the mean and an actual data point.
• Deviations can be calculated by taking each score and subtracting the mean from it: Deviance = Outcome - Model

Total Error

• The error between the mean and the data can be added together
• Deviations cancel out because some are positive and others are negative.
• To resolve this, each deviation should be squared.
• The Sum of Squared Errors (SS) is when squared deviations are added together.

Sum of Squared Errors

• The formula to find the Sum of Squared Errors is SS = Σ(X - X̄)²

Mean Squared Error

• The accuracy of the model can be measured by the SS, but it depends on the amount of data collected.
• To overcome this problem, we use mean squared error.

Degrees of Freedom

• Degrees of freedom allows the final score to be the value that makes the sample mean equal to 10.

Standard Error

• SD tells us how well the mean represents the sample data.
• If estimating the parameter in the population is desired, the standard error is needed.

SD and the Shape of a Distribution

• With a large SD, the width and spread of the data is larger.
• With a small SD, the width and spread of the data is smaller

Samples and Population

• The mean and SD describe only the sample from which they were calculated.
• For the population, the mean and SD are intended to describe the entire population, which is rare in psychology.
• For samples to populations, the mean and SD are obtained from a sample but are used to estimate the mean and SD of the population (very common in psychology).

Confidence Intervals and Statistical Significance

• This contains information about intervals.
• Some intervals do not contain the true value.

Confidence Intervals

• Domjan et al. (1998) studied conditioned sperm release in Japanese Quail.
• The true mean was 15 million sperm.
• The sample mean was 17 million sperm.
• The interval estimate was 12 to 22 million (contains true value).
• The interval estimate was 16 to 18 million (misses true value).
• Cls are constructed such that 95% contain the true value.

Types of Hypotheses

• The null hypothesis, H0, states that there is no effect.
• For example, Big Brother contestants and members of the public won't differ in their scores on personality disorder questionnaires.
• The alternative hypothesis, H1 (AKA the experimental hypothesis), states that Big Brother contestants will score higher on personality disorder questionnaires than members of the public.

Visual Representation of Hypotheses

• Alternative hypothesis leads to generating a testable prediction.
• The sample size needed (power) is calculated; specify a significance level, a.
• To sample randomly, computed sample statistic(s).
• Compute long-run probability, p, that the observed statistic would be at least as big as it is if the null hypothesis were true.
• Compare p to a.
• Should p be less than or equal to a, reject the null.
• Should p be greater than a, accept the null.

Test Statistics

• A statistic for which the frequency of particular values is known should be identified.
• Observed values can be used to test hypotheses.
• The equation used to test statistic = signal/noise = variance explained by the model/variance not explained by the model = effect/error

One- and Two- Tailed Tests

• Mean of group 1 is bigger than the mean of group 2, or there is a positive relationship.
• Mean of group 1 is smaller than the mean of group 2, or there is a negative relationship.

Type I and Type II Errors

• A Type I error occurs when believing that there is a genuine effect in our population, when in fact there isn't.
• The probability is the a-level (usually 0.05).
• A Type II error occurs when believing that there is no effect in the population, when in reality, there is.
• The probability is the B-level (often 0.2).