Aabel Statistics & Multivariate Data Analysis Methods

Many inferential tests makes certain assumptions about the shape of an underlying population distribution, and in this regard, the most commonly encountered assumption is that the underlying population from which each of the samples is derived is normal.

The shape of a normal distribution (also referred to as Gaussian distribution) is such that the closer a score is to the mean, the more frequently it occurs. The more extreme the deviation of a score from the mean, the less frequently it occurs.

Aabel provides:

• Probability plot and accompanying Shapiro-Wilk test
• Kolmogorov-Smirnov test for testing normality of a single sample
• Shapiro-Wilk test for normality

Homogeneity of variance is a reference to equal variances across groups/samples.

• Hartley's Fmax test
• Bartlett's chi-square test

These tests evaluate whether or not the population variances represented by k >= 2 samples/groups are equal.

Aabel provides the following ANOVA options:

• One-way between-subjects ANOVA:

  The test statistic evaluates if there is a significant difference between at least two of the group means in a set of k means; for more information, click here

• Two-way between-subjects ANOVA:

  The test statistic evaluates the effect of two independent variables (factors) on a response variable simultaneously. That is, it evaluates the variation among the differences between means for different levels of one factor over different levels of the other factor; for more information, click here

• One-way within-subjects (repeated measures) ANOVA:

  The test statistic evaluates if there is a significant difference between at least two of the conditions of repeated measures in a set of k conditions; Repeated measures ANOVA is used when all subjects of a random sample/group are measured under a number of different conditions; for more information, click here

• Two-way repeated measures ANOVA (within-subjects factorial) ANOVA:

  Two-way repeated measures ANOVA combines elements of two-factor between-subjects design and single-factor within-subjects design. That is, the variability is computed for the main effects of Factor A, the main effect of Factor B, and the interaction (AB), but the design requires three error terms, one for each effect; for more information, click here)

• Two-way mixed factorial ANOVA (mixed between-within design):

  Two-way mixed model ANOVA, also known as split-plot factorial design, combines one independent sample factor (Factor A) and one correlated groups factor (Factor B), evaluating the main effects of the Factor A, Factor B, and the interaction (AB); for more information, click here

Multiple comparisons/post-hoc tests accompanying the analysis of variance methods in Aabel include:

• Tukey's HSD Test
• Newman-Keuls (Neuman-Keuls) Test on Ordered Means
• Tukey B Test for Contrast on Ordered Means
• Dunnett Test
• Tukey-Kramer Test
• Scheffé Test
• Bonferroni-Dunn Test

For more information regarding multiple comparisons accompanying ANCOVA, click here.

This is an extension of single-factor between-subjects ANOVA. If some of the variations in the dependent variable scores are caused by the effect of another continuous variable (covariate), use of ANCOVA will remove this variation from the error or random variance, resulting in increased sensitivity of the test for treatment effects. The test output includes:

• Analysis of covariance, analysis of variance for dependent, and analysis of variance of covariate.
• Testing for homogeneity of regression
• Tukey's HSD Test on Adjusted Means
• Scheffé Test Adjusted Means

For more information regarding single-factor between-subjects ANCOVA, click here.

• The Chi-Square Goodness-of-Fit Test:

  This test, also referred to as the χ² test for a single sample, evaluates whether the observed cell frequencies are different from the expected cell frequencies.

• The Single-Sample Chi-Square Test for Population Variance:

  This test evaluates whether a sample of n subjects come from a parent population in which the variance equals a specific hypothesized value.

  For more information about chi-square tests, click here.

Aabel provides the following t-Tests:

• Single samples t-test:

  This test evaluates whether a sample of n observations comes from a parent population in which the mean equals a specific (hypothesized) value.

• Paired samples t-test:

  This test evaluates whether two dependent samples represent populations with same or different means. The two samples must have the same number of subjects/objects, and must be matched as nearly as possible to exclude the effects of extraneous variation.

• Unpaired samples t-test:

  This test evaluates whether two independent samples/groups represent two populations with different means, while assuming the populations having the same variances.

This is a parametric test that evaluates whether or not two populations from which the two sample are taken have equal variances (or equal standard deviations). The number of subjects/objects can be the same or different for the two samples.

Aabel provides the following z-Tests:

• Paired samples z-test:

  This test evaluates whether two dependent samples represent populations with different means while using the known variances of the populations to calculate the z value. The two samples must have the same number of data points, and must be matched as nearly as possible.

• Unpaired samples z-test:

  This test evaluates whether two independent samples/groups represent populations with different means while using the known variances of the populations to calculate the z value. The two samples can have the same or a different number of subjects/objects.

Aabel provides the following correlation methods:

• Correlations and covariance matrices:

  This method allows generating a correlation or covariance matrix from numeric data columns.

• Pearson product-moment correlation coefficient (Pearson's r):

  Pearson r is a measure of correlation between two variables. The associated tests evaluate whether the correlation coefficient for the underlying population is different from zero, i.e., whether there is a monotonous relationship between the two variables.

• Fisher's z transformation (z_r) (provided for both correlation matrix and Pearson's r):

  This option is provided to allow transforming a skewed sampling distribution into a normalized format.

• Spearman's rank-order correlation coefficient (Spearman's ρ) (non-parametric):

  Spearman's ρ is the non-parametric analog of the Pearson product-moment correlation coefficient. The results or the former and latter are closely similar, as the Spearman correlation is calculated in a very similar manner as Pearson, except that Spearman first ranks the data.

• Kendall's rank correlation coefficient (Kendall's τ) (non-parametric):

  Like Spearman's ρ, Kendall's τ is a non-parametric method of correlation between two variables, but has an advantage over Spearman's ρ: Kendall's τ also indicates the difference between the probability that the observed data are in the same order for the two variables vs. the probability that the observed data are in different orders for the two variables.

  For more information about correlation methods, click here.

Certain quantities of interest in medicine, psychology, etc., can not be measured explicitly. Accordingly, the assessment is approached by asking a series of questions and combining the answers into a single numerical value, or by a scale such as pass/fail, yes/no, or other dichotomous items.

To form a scale in this manner requires internal consistency, i.e., the items should all measure the same construct. Aabel provides the following methods for internal consistency reliability estimates:

• Cronbach's alpha
• Kuder-Richardson rho (Formula 20)
• Kuder-Richardson rho (Formula 21)

For more information, click here.

• Wilcoxon signed-ranks test:

  This test evaluates whether a sample of n observations is drawn from a population in which the median equals a specific (hypothesized) value.

• Wilcoxon matched pairs signed-ranks test:

  This test evaluates whether or not the median of the difference scores equals zero. The corresponding parametric test is the paired samples t-Test.

• Mann-Whitney U test (Wilcoxon ranks sum test):

  The Mann-Whitney test also known as the Wilcoxon ranks sum test and the Man-Whitney-Wilcoxon test, is the non-parametric analog of the unpaired samples t-test.

• Kruskal-Wallis test:

  he Kruskal-Wallis test is the non-parametric analog of a one-way ANOVA. The Kruskal-Wallis test is used to compare independent samples, and tests the hypothesis that several populations have the same continuous distribution, at least as far as their medians are concerned.

• Spearman's rank-order correlation coefficient:

  Spearman's ρ is the non-parametric analog of the Pearson product-moment correlation coefficient. The results or the former and latter are closely similar, as the Spearman correlation is calculated in a very similar manner as Pearson, except that Spearman first ranks the data.

• Kendall's rank correlation coefficient (Kendall's tau):

  Like Spearman's ρ, Kendall's τ is a non-parametric method of correlation between two variables, but has an advantage over Spearman's ρ: Kendall's τ also indicates the difference between the probability that the observed data are in the same order for the two variables vs. the probability that the observed data are in different orders for the two variables.

• Friedman two-way analysis of variance by ranks:

  This test is the non-parametric version of the repeated measures ANOVA, and evaluates the hypothesis that several populations have the same continuous distribution, at least as far as the medians are concerned.

• Kolmogorov-Smirnov goodness-of-fit test for a single sample:
  This test is used to evaluate whether or not the distribution of data in a sample conform to a normal distribution with a specific hypothesized mean and standard deviation. The Kolmogorov-Smirnov goodness-of-fit test for a single sample is a test of ordinal data, because it requires that a cumulative frequency distribution be constructed (i.e., the scores need to be arranged in order of magnitude).

  For more information about non-Parametric tests, click here.

• Two-way contingency tables (the chi-square test of independence for r x c tables:

  A contingency table is a two-way frequency table used to summarize two categorical random variables. This method can provide two-way tables for observed frequencies, expected frequencies, and their differences. The chi-square test of independence for r x c tables is a non-parametric test for evaluating whether or not the two dimensions are independent at the specified significance level.

• Two-way tables of cell descriptive statistics:

  When using the contingency tables UI controls, in addition to or instead of frequency tables, you can generate two-way tables of cell descriptive statistics, i.e., mean, median, variance, standard variation, standard error, etc.

• Mosaic/Parquet matrix diagrams:

  The mosaic plot is a graphical method for visualizing n x m contingency tables. The areas of the mosaic tiles are proportional to the observed frequency of groups in the contingency table. In a parquet plot, the area of each rectangle is proportional to the expected frequency of groups in the contingency table. Aabel allows color coding the tiles based on (a) several statistical models, (b) observed and expected frequencies to display the quantities, and (c) rows and columns.

  For more information and examples of mosaic plots, click here.

Aabel provides methods for Kaplan-Meier survival analysis using either raw survival data or summarized survival data.

If the Kaplan-Meier curve is plotted without taking into account the number of subjects remaining at risk beyond completion of the study, the shape of the curve will be unaffected, but the survival probability values will be affected.

• Some raw data include the subjects at risk beyond completion of the study1 as additional rows in the data table (these added rows are tagged with a status code different from the event or censor code). Others may not include the subjects at risk beyond completion of the study. If your data is of the latter type, but you know the number of subjects at time zero, Aabel provides a flexible means of including the information in the analysis.

The Kaplan-Meier survival analysis output includes:

• Survival curves (default output) comparing the cumulative probability of survival at any specific time, with the option of displaying the censored observations
• Life table computation results
• Logrank significance test report - When we have generated two or more survival curves, the logrank test report is used to determine whether the differences in survival between groups, treatments, etc., are more than we would expect by chance alone.

The logrank is a non-parametric test that makes use of the full survival data without making any assumption about the shape of the survival curve or the distribution of survival times. The logrank test results include the hazard Ratio (i.e., the risk factor for one group, treatment, etc. compared to another group, treatment. etc.), the logrank chi-square and p values

For more information about Kaplan-Meier survival analysis and the accompanying log-rank test, click here.

ROC curves were originally developed to analyze noisy radio signals. Today, however ROC curves are widely used to display a plot of the true positive rate against the false positive rate for the different possible cut-points of a diagnostic test.

Aabel's implementation of ROC allows plotting:

• ROC for a single test
• ROC for two tests on paired samples
• ROC for two tests on unpaired samples

For more information, see examples.

The regression methods grouped under this title either deal with finding the relationship between one outcome (dependent) variable and one predictor (independent) variable or finding the relationship between two variables where the designation of dependent and independent variables is irrelevant.

Pre-defined regression including:

• Linear (X on Y)
• Linear (thru zero)
• Major axis
• Reduced major axis
• Polynomial
• Exponential
• Logarithmic
• Power

For more information and examples, click here.

Cubic Spline interpolation:

• A cubic spline is made from piece-wise third-order polynomials which pass through the control points provided. You can apply the cubic spline interpolation to a graph, or generate a specified sequence of interpolations for other uses.

User-Defined Non-Linear Regression

• User-define regression provides a library of functions and an interactive interface.

For more information regarding regression methods in Aabel, click here.

Logistic regression allows you to predict a discrete outcome from a set of independent variables that may be continuous, discrete, or binary. The dependent variable is binary/dichotomous/binominal. Logistic regression in Aabel includes probability and logit (probability). Aabel allows generating:

• Probability plots with one independent variable
• Probability plots with multiple dimension projections

Logistic regression statistics (for probability plots with multiple dimension projections) includes logistic regression full model summary report and logistic regression parameters summary report.

For more information and examples, click here.

• Predicted values and residuals
• The regression summary table (providing regression parameters, the corresponding test statistic, and the confidence limits)
• ANOVA report for total regression
• Partial regression ANOVA report - The test results for the significance of predictor variables

For more information, click here.

For more information, click here.

PCA is a data reduction method with the primary objective of extracting a few uncorrelated variables that may capture most of the variability in a data set, while preserving the orthogonality of these new optimal reference axes/variables (i.e., principal components). The 1st principal component captures the maximum variation in the data set. The 2nd principal component has the next most variation, and so on.

• PCA analysis allows optional pre-processing of the data prior to the main analysis; the options includes:
  • Standardizing
  • Normalizing
  • Logarithmisizing
  • Log centering
  • Mean centering
  • Taking square root
  • Ranking variables individually
  • Ranking variables jointly
• PCA results can be stored in the source worksheet or in a new worksheet.
• The scores and loadings can be graphically displayed using Aabel graphs and tables.

For more information, click here.

The defining characteristic that distinguishes between PCA and factor analysis is that in PCA we assume that all variability in an item should be used in the analysis, while in factor analysis, we define a priory the number of factors that we want to extract, and the extracted axes will be scaled to the variance along the new improved axes.

• Factor analysis allows optional pre-processing of the data prior to the main analysis; the options includes:
  • Logarithmisizing
  • Log centering
  • Mean centering
  • Taking square root
  • Ranking variables individually
  • Ranking variables jointly
• Aabel provides R-mode and Q-mode factor analyses and the option for Kaiser varimax rotation.
• The loadings, communality, and "unique" data will be placed in an auto-generated worksheet.

You can use Aabel graphing capabilities to display the loadings, the fraction of variance of the variables explained by the model (i.e., communality) and the fraction that is not (i.e., "unique").

For more information, click here.

Aabel provides the following outlier analysis methods:

• Mahalanobis Distance (a widely used method of multivariate outlier detection)
• Jackknifed Mahalanobis Distance

Log Center transformation is necessary for proportion or percentage data, i.e., for data where the row sums of the selected variables have a constant sum. The outlier analysis UI controls include the necessary pre-processing transformation methods (such as log-centering), which can be chosen, if needed.

For more information, click here.

This multivariate method is used for clustering/grouping data with similar characteristics.

• The method partition a data set of n objects into k clusters via an iterative process that continues until the sum of squares from points to the assigned cluster centers is minimized, i.e., until all cluster centers are at the mean of their voronoi sets.
• Aabel uses the method of Hartigan and Wong (1979), performs several random starts, and attempts to converge to a global minimum of the squared error distortion.

For graphical representation of k-means analysis, you can use a 2-D (binary or matrix) or 3-D plot that allows displaying scatter data points; members of each cluster, which have similar characteristics in multidimensional space, will be displayed with identical marker properties.

Aabel allows optional pre-processing of the data prior to the main k-means clustering analysis. The options include:

• Standardizing
• Normalizing
• Logarithmisizing
• Log centering
• Mean centering
• Taking square root
• Ranking variables individually
• Ranking variables jointly

For more information, click here.

The method used in Aabel is called the weighted pair-group with arithmetic averaging. When objects/observations are defined by a set of numeric variables, each object (worksheet row) is positioned in a multi-dimensional space of a dimension proportional to the number of variables (worksheet columns) used to define the object.

The dissimilarity/similarity measures in Aabel are based on one of the following options:

• The correlation coefficient (Pearson's product moment correlation coefficient)
• Euclidean distance similarity (straight line)
• Standardized Euclidian distance coefficient
• Manhattan distance similarity measure

For an example, click here.

Control charts are used to monitor a process for some quality characteristic such as e.g., thickness, weight, defective fractions etc.

Shewhart Control Charts for Variables: These charts are based on quality characteristics that can be measured and expressed numerically:

• Xbar (R) chart
• Xbar (S) chart
• R chart
• S chart

Individual Measurements and Moving Range: Control charts for individual measurements use the moving range of two successive observations to measure the process variability.

• Individual measurements chart
• Moving range chart

Dot plots are an alternative to histograms of continuous data; in a dot plot, each data point (individual observation) is plotted on a continuous scale using a symbol (on the X-axis). Aabel applies the method published by Leland Wilkinson (1999) to generate dot plots, but in addition, uses modifications necessary to allow:

• Dynamically compensating for modifying the plot aspect ratio
• Scaling of dot size (width)
• Using ellipsoids dot stacks for making the plot more readable when dot stacks are either too uniform or too cluttered by vertical overlaps
• Option of applying object marker colors to dot symbols allows distinguishing groups of data.

For more information and examples, click here.

This graph is designed to display dot plots of score (response values) form k >=2 repeated measures (dependent samples) on axes that are parallel to one another and equally spaced, with all axes having the same value range (see the right-hand side image).

• Lines extending from their positions on one axis to the next connect the dependent data points.
• The probability significance test (p value) provided with the plot is based on one-way repeated measures ANOVA.

For more information and an example, click here.

The box & whisker graphs display rank statistics. They can have the form of a box or notched box that spans the distance between the two quartiles surrounding the median.

These plots are used for comparing mean, medians, maximum, or minimum values of multiple variables, or of subgroups/categories of variables.

Aabel provides the following graph types:

• One-way bar plots
• Two-way bar plots
• One-way line plots
• Two-way line plots

  To add error bars to mean bars/lines, the options include:

  • Standard error of mean
  • Standard deviation
  • Confidence interval (including options of 90.0%, 95.0%, 97.5%, 99.0%)

For more information regarding bar plots, click here; for more information regarding line plots, click here.

Interaction plots are stacks of mean lines, used to display the effect of one factor at each level of another factor.

With no or insignificant interaction, the mean lines are approximately parallel. The more the lines deviate from being parallel, the more significant the interaction effect.

To add error bars to interaction plots, you can choose one of the following options:

• Standard error of mean
• Standard deviation
• Confidence interval (including options of 90.0%, 95.0%, 97.5%, 99.0%)

For more information and an example graph, click here.

These plots compare the response values (scores, measurements) obtained from k >=2 samples/groups, each of which representing data from pqs levels of experimental conditions. The plot options include:

• Three-way mean bar plot
• Three-way mean dot plot
• To add error bars to mean bars/dots, you can choose one of the following options:
  • Standard error of mean
  • Standard deviation
  • Confidence interval (including options of 90.0%, 95.0%, 97.5%, 99.0%)

For more information and example plots, click here.

In these plots, the horizontal dashed line is the overall mean. The line through the center of each diamond is the group mean. The top and bottom diamond vertices are the respective upper and lower 95% confidence limits (CI) about the group mean.

In groups with equal sample sizes, overlapping marks indicate that the two group means are not significantly different at the 95% confidence level.

Aabel provides the following diamond plot types:

• One-way diamond mean comparison plot
• Two-way diamond mean comparison plot

For more information, click here.

The Bland & Altman method comparison compares two methods of measurement or two paired variables, and provides a plot of difference vs. mean, in which the standard deviation of differences between measurements made by the two methods provides an index of the comparability of the methods.

In Aabel, the Bland & Altman method provides:

• A plot of differences vs. mean
• A plot of differences as a % of averages vs. mean
• A plot of ratios vs. mean
• In addition to the plot, the results can optionally be displayed in a table or stored in a worksheet.
• Furthermore, Aabel has the option of showing:
  • The ±1.96 standard deviation, which is commonly used in a typical Bland & Altman plot
  • A paired t-test difference plot with the 95% CI of mean difference above and below the mean line
  • A paired t-test difference plot with the 99% CI of mean difference above and below the mean line

For example graphs, click here.