What is a factorial ANOVA?

A factorial ANOVA is any ANOVA that uses more than one categorical independent variable. A two-way ANOVA is a type of factorial ANOVA.

Some examples of factorial ANOVAs include:

Testing the combined effects of vaccination (vaccinated or not vaccinated) and health status (healthy or pre-existing condition) on the rate of flu infection in a population.
Testing the effects of marital status (married, single, divorced, widowed), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.
Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded, somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.

Frequently asked questions: Statistics

Are ordinal variables categorical or quantitative?: In statistics, ordinal and nominal variables are both considered categorical variables.

Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them.
What’s the difference between central tendency and variability?: While central tendency tells you where most of your data points lie, variability summarizes how far apart your points from each other.

Data sets can have the same central tendency but different levels of variability or vice versa. Together, they give you a complete picture of your data.
What are the 4 main measures of variability?: Variability is most commonly measured with the following descriptive statistics:

Range: the difference between the highest and lowest values

Interquartile range: the range of the middle half of a distribution

Standard deviation: average distance from the mean

Variance: average of squared distances from the mean
What is variability?: Variability tells you how far apart points lie from each other and from the center of a distribution or a data set.

Variability is also referred to as spread, scatter or dispersion.
What is the difference between interval and ratio data?: While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.

For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.
What is a critical value?: A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval, or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.
What is the difference between the t-distribution and the standard normal distribution?: The t-distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z-distribution).

In this way, the t-distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance, you will need to include a wider range of the data.
What is a t-score?: A t-score (a.k.a. a t-value) is equivalent to the number of standard deviations away from the mean of the t-distribution.

The t-score is the test statistic used in t-tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t-distribution.
What is a t-distribution?: The t-distribution is a way of describing a set of observations where most observations fall close to the mean, and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

The t-distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation.
What is the range in statistics?: In statistics, the range is the spread of your data from the lowest to the highest value in the distribution. It is the simplest measure of variability.
What is ordinal data?: Ordinal data has two characteristics:

The data can be classified into different categories within a variable.

The categories have a natural ranked order.

However, unlike with interval data, the distances between the categories are uneven or unknown.
What’s the difference between nominal and ordinal data?: Nominal and ordinal are two of the four levels of measurement. Nominal level data can only be classified, while ordinal level data can be classified and ordered.
What does it mean if my confidence interval includes zero?: If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.

If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data.

In both of these cases, you will also find a high p-value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.
How do I calculate a confidence interval if my data are not normally distributed?: If you want to calculate a confidence interval around the mean of data that is not normally distributed, you have two choices:

Find a distribution that matches the shape of your data and use that distribution to calculate the confidence interval.

Perform a transformation on your data to make it fit a normal distribution, and then find the confidence interval for the transformed data.
What is a standard normal distribution?: The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be converted into the standard normal distribution by turning the individual values into z-scores. In a z-distribution, z-scores tell you how many standard deviations away from the mean each value lies.
What are z-scores and t-scores?: The z-score and t-score (aka z-value and t-value) show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z-distribution or a t-distribution.

These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z-score of 2.5, this means that your estimate is 2.5 standard deviations from the predicted mean.

The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.
How do you calculate a confidence interval?: To calculate the confidence interval, you need to know:

The point estimate you are constructing the confidence interval for

The critical values for the test statistic

The standard deviation of the sample

The sample size

Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate (e.g. a mean or a proportion) and on the distribution of your data.
What is the difference between a confidence interval and a confidence level?: The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.

The confidence interval is the actual upper and lower bounds of the estimate you expect to find at a given level of confidence.

For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find an upper bound of 0.56 and a lower bound of 0.48. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.

This means that 95% of the time, you can expect your estimate to fall between 0.56 and 0.48.
What’s the difference between the range and interquartile range?: While the range gives you the spread of the whole data set, the interquartile range gives you the spread of the middle half of a data set.
What are the different types of means?: The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:

Weighted mean: some values contribute more to the mean than others.

Geometric mean: values are multiplied rather than summed up.

Harmonic mean: reciprocals of values are used instead of the values themselves.
How do I find the mean?: You can find the mean, or average, of a data set in two simple steps:

Find the sum of the values by adding them all up.

Divide the sum by the number of values in the data set.

This method is the same whether you are dealing with sample or population data or positive or negative numbers.
When should I use the median?: The median is the most informative measure of central tendency for skewed distributions or distributions with outliers. For example, the median is often used as a measure of central tendency for income distributions, which are generally highly skewed.

Because the median only uses one or two values, it’s unaffected by extreme outliers or non-symmetric distributions of scores. In contrast, the mean and mode can vary in skewed distributions.
How do I find the median?: To find the median, first order your data. Then calculate the middle position based on n, the number of values in your data set.

If n is an odd number, the median lies at the position (n + 1) / 2.

If n is an even number, the median is the mean of the values at positions n / 2 and (n / 2) + 1.
Can there be more than one mode?: A data set can often have no mode, one mode or more than one mode – it all depends on how many different values repeat most frequently.

Your data can be:

without any mode

unimodal, with one mode,

bimodal, with two modes,

trimodal, with three modes, or

multimodal, with four or more modes.
How do I find the mode?: To find the mode:

If your data is numerical or quantitative, order the values from low to high.

If it is categorical, sort the values by group, in any order.

Then you simply need to identify the most frequently occurring value.
When should I use the interquartile range?: The interquartile range is the best measure of variability for skewed distributions or data sets with outliers. Because it’s based on values that come from the middle half of the distribution, it’s unlikely to be influenced by outliers.
What are the two main methods for calculating interquartile range?: The two most common methods for calculating interquartile range are the exclusive and inclusive methods.

The exclusive method excludes the median when identifying Q1 and Q3, while the inclusive method includes the median as a value in the data set in identifying the quartiles.

For each of these methods, you’ll need different procedures for finding the median, Q1 and Q3 depending on whether your sample size is even- or odd-numbered. The exclusive method works best for even-numbered sample sizes, while the inclusive method is often used with odd-numbered sample sizes.
What is nominal data?: Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way.

For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.
What is homoscedasticity?: Homoscedasticity, or homogeneity of variances, is an assumption of equal or similar variances in different groups being compared.

This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. Uneven variances in samples result in biased and skewed test results.
What is variance used for in statistics?: Statistical tests such as variance tests or the analysis of variance (ANOVA) use sample variance to assess group differences of populations. They use the variances of the samples to assess whether the populations they come from significantly differ from each other.
What’s the difference between standard deviation and variance?: Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in a distribution, but their units differ:

Standard deviation is expressed in the same units as the original values (e.g., minutes or meters).

Variance is expressed in much larger units (e.g., meters squared).

Although the units of variance are harder to intuitively understand, variance is important in statistical tests.
What is the empirical rule?: The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution:

Around 68% of values are within 2 standard deviations of the mean.

Around 95% of values are within 4 standard deviations of the mean.

Around 99.7% of values are within 6 standard deviations of the mean.

The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.
What is a normal distribution?: In a normal distribution, data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.

The measures of central tendency (mean, mode and median) are exactly the same in a normal distribution.
What does standard deviation tell you?: The standard deviation is the average amount of variability in your data set. It tells you, on average, how far each score lies from the mean.

In normal distributions, a high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean.
Can the range be a negative number?: No. Because the range formula subtracts the lowest number from the highest number, the range is always zero or a positive number.
What is simple linear regression?: Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.

For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.
What is a model?: In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable.

You can test a model using a statistical test. To compare how well different models fit your data, you can use Akaike’s information criterion for model selection.
How is AIC calculated?: The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters (K) used to reach that likelihood. The AIC function is 2K – 2(log-likelihood).

Lower AIC values indicate a better-fit model, and a model with a delta-AIC (the difference between the two AIC values being compared) of more than -2 is considered significantly better than the model it is being compared to.
What is the Akaike information criterion?: The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe. It penalizes models which use more independent variables (parameters) as a way to avoid over-fitting.

AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data.
How is statistical significance calculated in an ANOVA?: In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.

Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).

If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.
What is the difference between a one-way and a two-way ANOVA?: The only difference between one-way and two-way ANOVA is the number of independent variables. A one-way ANOVA has one independent variable, while a two-way ANOVA has two.

One-way ANOVA: Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.

Two-way ANOVA: Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior, master’s), and race finishing times in a marathon.

All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.
What is multiple linear regression?: Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.
How is the error calculated in a linear regression model?: Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

measuring the distance of the observed y-values from the predicted y-values at each value of x;

squaring each of these distances;

calculating the mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.
What is meant by model selection?: In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data.

The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision.

AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting.
What is a regression model?: A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.
Can I use a t-test to measure the difference among several groups?: A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.
What is the difference between a one-sample t-test and a paired t-test?: A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).
What does a t-test measure?: A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).
Which t-test should I use?: Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test.

If you want to know only whether a difference exists, use a two-tailed test. If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test.
What is a t-test?: A t-test is a statistical test that compares the means of two samples. It is used in hypothesis testing, with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.
What is statistical significance?: Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test. Significance is usually denoted by a p-value, or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis.

When the p-value falls below the chosen alpha value, then we say the result of the test is statistically significant.
What is a test statistic?: A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean, or how different a linear slope is from the slope predicted by a null hypothesis. Different test statistics are used in different statistical tests.
How do you calculate a p-value?: P-values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p-value tables for the relevant test statistic.

P-values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.

If the test statistic is far from the mean of the null distribution, then the p-value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.
What’s the best measure of central tendency to use?: The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.
Which measures of central tendency can I use?: The measures of central tendency you can use depends on the level of measurement of your data.

For a nominal level, you can only use the mode to find the most frequent value.

For an ordinal level or ranked data, you can also use the median to find the value in the middle of your data set.

For interval or ratio levels, in addition to the mode and median, you can use the mean to find the average value.
What are measures of central tendency?: Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

The mode is the most frequent value.

The median is the middle number in an ordered data set.

The mean is the sum of all values divided by the total number of values.
How do I decide which level of measurement to use?: Some variables have fixed levels. For example, gender and ethnicity are always nominal level data because they cannot be ranked.

However, for other variables, you can choose the level of measurement. For example, income is a variable that can be recorded on an ordinal or a ratio scale:

At an ordinal level, you could create 5 income groupings and code the incomes that fall within them from 1–5.

At a ratio level, you would record exact numbers for income.

If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is.
Why do levels of measurement matter?: The level at which you measure a variable determines how you can analyze your data.

Depending on the level of measurement, you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis.
What are the four levels of measurement?: Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:

Nominal: the data can only be categorized.

Ordinal: the data can be categorized and ranked.

Interval: the data can be categorized and ranked, and evenly spaced.

Ratio: the data can be categorized, ranked, evenly spaced and has a natural zero.
Does a p-value tell you whether your alternative hypothesis is true?: No. The p-value only tells you how likely the data you have observed is to have occurred under the null hypothesis.

If the p-value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.
Which alpha value should I use?: The alpha value, or the threshold for statistical significance, is arbitrary – which value you use depends on your field of study.

In most cases, researchers use an alpha of 0.05, which means that there is a less than 5% chance that the data being tested could have occurred under the null hypothesis.
What are the main assumptions of statistical tests?: Statistical tests commonly assume that:

the data are normally distributed

the groups that are being compared have similar variance

the data are independent

If your data does not meet these assumptions you might still be able to use a nonparametric statistical test, which have fewer requirements but also make weaker inferences.
What is a p-value?: A p-value, or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test.
How do I know which test statistic to use?: The test statistic you use will be determined by the statistical test.

You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.
What factors affect the test statistic?: The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis, even if the true correlation between two variables is the same in either data set.
How do you calculate a test statistic?: The formula for the test statistic depends on the statistical test being used.

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation).
What’s the difference between univariate, bivariate and multivariate descriptive statistics?: Univariate statistics summarize only one variable at a time.

Bivariate statistics compare two variables.

Multivariate statistics compare more than two variables.
What are the 3 main types of descriptive statistics?: The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.

Distribution refers to the frequencies of different responses.

Measures of central tendency give you the average for each response.

Measures of variability show you the spread or dispersion of your dataset.
What’s the difference between descriptive and inferential statistics?: Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

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