Part of a statistical hypothesis. There is no specific alternate hypothesis.

Thing 1 ≠ Thing 2

Thing 1 > Thing 2

A test for normality. Focuses on deviations in tails.

A² = (average of differences)²

Analysis of variance

The data must have a Gaussian distribution to use the ANOVA test.

A goodness of fit test for continuous data. the most common type is the Pearson χ², which is sensitive; if you change one value by even a small amount, it can dramatically alter the end results. Corrections include the Yates continuity correction, maximum likelihood of chi-square, and Fisher's exact test. Evaluates whether the observed cell frequencies are different than expected cell frequencies. The smaller the value, the closer the observed and expected frequencies are, and the better the fit. If the value is zero, the observed and expected frequencies are equal. The p-value associated with the statistic and the degrees of freedom may be found in a chart, or calculated on SAS. The null hypothesis is that the observed frequency and expected frequency are the same.

χ² = Σ((observed - expected)² / expected)

An experimental design. Treatments are randomly assigned to experimental units. Variation is partitioned into one source: the treatments. Uses a fixed effects model; treatment effect is fixed. Can used the F-test ratio. A good design if experimental units are homogenous. No control over the experimental error variance. Where Y_ij is the observation on the jth experimental unit of the ith treatment, μ is the overall mean, t_i is the ith treatment effect, and e_ij is the random error of the jth experimental unit on the ith treatment:

Y_ij = μ + t_i + e_ij

Crosstab table

A table containing two or more categorical variables. We want to examine the variables at the same time. May be frequency-by-frequency. If there are a lot of variables the table may be difficult to display.

Interval data

Ratio data

Items that are measured with a continuous scale. You can calculate the average and it has meaning. You can use a test for normality to see if the sample is representative of the larger population. Examples include weight, height, age, temperature, etc.

A test for normality.

W² = (sum of differences)²

Used to find the correct critical value of a t-statistic in a table. The number of groups minus one.

df = (n₁ + n₂) - 2

Categorical data

Includes nominal and ordinal data. Each datapoint can belong to only one group or category. You may be able to calculate a mean, but it will have no meaning. Cannot be used in a test for normality. May be used in an odds ratio. Typically has a Poisson distribution.

Based on the research question. Drives what data to collect and analyse. The process of planning a study to meet specific objectives. The experiment should be designed to match a specified research question. Experimental designs include CRD, RCBD, split-plot, strip-plot, and repeated measures. Steps include:

1. Define experimental unit

2. Identify types of variables you are collecting

3. Define the treatment structure

4. Define the overall structure

Normal distribution.

A classic bell curve. Described by the mean of the population (μ) and sample (y bar), and variance of the population (σ²) and sample (s²). The mean, median, and mode are equal. A perfectly symmetrical distribution. When testing the difference between two treatments, you do not want the data to be normal; you want the treatment to affect the population. You should always check to see if residuals are normally distributed using a test for normality. Normality is subjective. 

Mean ± 1 σ = 68.27% of population

Mean ± 1.96 σ = 95% of population

Mean ± 2 σ = 95.44% of population

Mean ± 3 σ = 99.73% of population

A test for normality. Focuses on deviations near the centre.

D = largest difference between EDF and normal

An approach used as a crude, inexact way to quickly determine whether reported differences are different or not. The null hypothesis should be accepted if the range of the data is less than the LSD. Developed by R.A. Fisher. Can only be used when there are two treatments.

LSD = (critical t-statistic)(sed)

If n is the same in both groups,

LSD = (critical t-statistic)(√2)(sem) ≅ 3 se

Invented by Fisher. Accounts for the total variation: treatments, treatment interactions, design, covariance, and unknown. The unknown component is random error. Experimental error is split into treatments, treatment interactions, experimental design, covariances, and unknown error. We can account for the variation of all components except unknown error. Solved using two steps:

1. Fixed effects are solved

2. Random effects are solved

A correction for the chi-square test. Converts the Yates continuity correction into log scale.

χ2 = 2(Σ((observed)(ln(observed / expected))))

Part of a statistical hypothesis. The decision is always about the null hypothesis; it is either accepted or rejected, with no condition or adjective added to qualify the hypothesis or effect size. If you reject the null hypothesis, there is a difference, and the alternate hypothesis is accepted. If you accept the null hypothesis, there is no difference.

Thing 1 = Thing 2

The probability of correctly rejecting the null hypothesis when it is false. Each comparison in an experiment has its own power. There is no single power for an analysis. Rate is based on the comparison. Related to sample size, effect size, variation of the outcome variable, and the p-value. To increase the power of the test, you can increase sample size (n) or reduce experimental error (σ). The effect size cannot be changed by the experimenter, however when the difference is greater, the power of the test increases. Usually we try to attain a power of 0.8, so that 80% of the time we are confident that the test will correctly reject the null hypothesis when it is false.

λ = ((μ₁ - μ₂)√n) / σ

"The use of inferential statistics to test for treatment effects with data from experiments where the treatments are not replicated (although samples may be) or replicates are not statistically independent" - Hurlbert (1984).

When there is only one experimental unit per treatment.

Gives more control over the experimental error variance by grouping experimental units into homogenous groups. The simplest blocking design used to control and reduce experimental error. The experimental units are grouped into blocks of homogenous units. Each treatment is randomly assigned an equal number of experimental units in each block.

Y_ij = μ + Trmt_i + Block_j + e_ij

A type of standard error. Used in calculating the t-statistic.

sed = √(s²((1 / n₁) + (1 / n₂)))

If n is the same in both groups,

sed = √(2s² / n) = (√2)(se)

S, specific

M, measurable

A, attainable and achievable

R, realistic, relevant, and reasonable

T, time-based, timely, and tangible

Relates to a population and its distribution. Always larger than standard error. Calculated when you have a large dataset that represents the whole population.

sd = √(s²)

Se of the mean (SEM). Relates to a statistic of a sample, such as a mean, and its distribution. Includes standard error of the mean (SEM). The measurement is one of many possible measurements from the population.

se = √(s² / n)

Student's t-distribution

A test to compare means. Developed by W.S. Gosset in 1906 with funding from Guinness. It was developed to select consistent samples of barley for use in brewing beer. First published in Biometrika in March 1908. Used to test small samples. The ratio of "how big" and "how confident'. The bigger the difference, and/or greater the confidence (lower se), the larger the t-statistic will be. Assumes that variation is the same between both means; error variance is the same. Historically, the value of the t-statistic was compared with its critical value from a table, based on the degrees of freedom. Today, software packages provide the calculated p-value. Not a good test for complex research models. Can only be used if there are two treatments. The null hypothesis is that the means are equal. 

t-statistic = (difference of means)/(sed)

Time-based information

Adds a dimension to any data collected, whether it be qualitative or quantitative in nature. Examples include weights taken every month, thoughts recorded every hour, etc.

False positive

When you reject the null hypothesis when ti is true. In actuality Thing 1 and Thing 2 are the same. Its probability is α.

False negative

When you accept the null hypothesis when it is false. In actuality Thing 1 and Thing 2 are different. Its probability is β, related to the power of the test.

A correction for the chi-square test that makes it less sensitive to small changes in data.

χ² = Σ((|observed - expected| - 0.05)² / expected)