Methods and formulas for the effects plots in Analyze Factorial Design

Normal probability plot

The method that Minitab uses to draw the normal effects plot depends on the degrees of freedom for the error term.

If the error term has one or more degrees of freedom, Minitab plots the normal scores, probabilities or percentages versus the standardized effects. The line corresponds to a normal distribution with a standard deviation of 1. Effects with p-values less than α are labeled significant on the graph. Minitab labels this graph Normal Plot of the Standardized Effects.

If the error term has zero degrees of freedom, Minitab plots the normal scores, probabilities or percentages versus the nonstandardized effects. The line corresponds to a normal distribution with a standard deviation given by Lenth's pseudo standard error. Effects with an absolute value exceeding the margin of error (ME) are labeled significant on the plot. Margin of error is:

  • ME = t*PSE

where t is the (1 – α / 2) quantile of a t-distribution with degrees of freedom equal to the (number of effects / 3). Minitab labels this graph Normal Plot of the Effects.

For more information on how Minitab calculates PSE, go to the section on Lenth's pseudo standard error.

Half normal plot

The method that Minitab uses to draw the half normal effects plot depends on the degrees of freedom for the error term.

If the error term has one or more degrees of freedom, Minitab plots the normal scores, probabilities or percentages versus the standardized effects. The line corresponds to a normal distribution with a standard deviation of 1. Effects with p-values less than α are labeled significant on the graph. Minitab labels this graph Half Normal Plot of the Standardized Effects.

If the error term has zero degrees of freedom, Minitab plots the normal scores, probabilities or percentages versus the nonstandardized effects. The line corresponds to a normal distribution with a standard deviation given by Lenth's pseudo standard error. Effects with an absolute value exceeding the margin of error (ME) are labeled significant on the plot. Margin of error is:

  • ME = t*PSE

where t is the (1 – α / 2) quantile of a t-distribution with degrees of freedom equal to the (number of effects / 3).

The half normal effects plot uses half normal plot points, which are based on the distribution of the absolute value of a standard normal random variable. The cumulative distribution function associated with the half normal plot is:

  • F(x) = 2Φ(x) – 1

where Φ is the cumulative distribution function for the standard normal distribution.

For more information on how Minitab calculates PSE, go to the section on Lenth's pseudo standard error.

Pareto chart

The method that Minitab uses to draw the Pareto chart of the effects depends on the degrees of freedom for the error term.

If the error term has one or more degrees of freedom, the red line on the Pareto chart is drawn at t, where t is the (1 – α / 2) quantile of a t-distribution with degrees of freedom equal to the degrees of freedom for the error term. Minitab labels this graph Pareto Chart of the Standardized Effects.

If the error term has zero degrees of freedom, Minitab identifies important effects using Lenth's pseudo standard error(PSE). The red line of the Pareto chart is drawn at the margin of error, which is:

  • ME = t*PSE

where t is the (1 – α / 2) quantile of a t-distribution with degrees of freedom equal to the (number of effects / 3). Minitab labels this graph Pareto Chart of the Effects.

For more information on how Minitab calculates PSE, go to the section on Lenth's pseudo standard error.

Lenth's pseudo standard error (PSE)

Lenth's pseudo standard error (PSE) is based on the concept of sparse effects, which assumes the variation in the smallest effects is due to random error. Minitab calculates PSE using the following method:

  1. Calculates the absolute value of the effects
  2. Calculates S, which is 1.5 * median of the effects in Step 1
  3. Calculates the median of the effects that are less than 2.5 * S
  4. Calculates PSE, which is 1.5 * the median calculated in Step 3
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