for Graphing and Data Visualization

With an award-winning interface and intuitive wizard technology that guides users step-by-step through the graph creation and data analysis process, SigmaPlot provides the flexibility to create compelling graphs and statistical analysis you simply can’t achieve with basic spreadsheet software.

## Overview

#### SigmaPlot Helps You Quickly Create Exact Graphs

With the new Graph Properties user interface you can select the property category in the tree on the left and then change properties on the right. The change is immediately graphed and if you move your cursor off the panel then it becomes transparent and you can see the effect of your changes without leaving the panel.

The “select left and change right” procedure makes editing your graphs quick and easy. SigmaPlot takes you beyond simple spreadsheets to help you show off your work clearly and precisely. With SigmaPlot, you can produce high-quality graphs without spending hours in front of a computer. SigmaPlot offers seamless Microsoft Office^{®} integration, so you can easily access data from Microsoft Excel^{®} spreadsheets and present your results in Microsoft PowerPoint^{®} presentations.

The user interface also includes Microsoft Office style ribbon controls. And the tabbed window interface efficiently organizes your worksheets and graphs for easy selection. And these tabs may be organized into either vertical or horizontal tab groups. Graph Gallery and Notebook Manger panes may be moved to any position and easily placed using docking panel guides. You can add frequently used objects to the Quick Access Toolbar. For example you might want to add Notebook Save, Close All, Refresh Graph Page and Modify Plot.

#### More than 100 2-D and 3-D technical graph types

From simple 2-D scatter plots to compelling contour and the new radar and dot density plots, SigmaPlot gives you the exact technical graph type you need for your demanding research. And, to help you see interactions in your 3-D data, SigmaPlot powerfully renders multiple intersecting 3-D meshes with hidden line removal. With so many different chart and graph types to choose from, you can always find the best visual representation of your data.

#### Use Global Curve Fitting to simultaneously analyze multiple data

Global curve fitting is used when you want to fit an equation to several data sets simultaneously. The selected equation must have exactly one independent variable. The data sets can be selected from a worksheet or a graph using a variety of data formats. You can also specify the behavior of each equation parameter with respect to the data sets. A parameter can be localized to have a separate value for each data set, or a parameter can be shared to have the same value for all data sets.

The Global Curve Fit Wizard is very similar to the Regression and Dynamic Fit Wizards in design and operation. The main difference is the extra panel shown below for specifying the shared parameters.

#### Obtain Data from Nearly Any Source

SigmaPlot has import file formats for all common text files. This includes a general purpose ASCII file importer which allows importing comma delimited files and user-selected delimiters. Plus all Excel formats may be imported. SPSS, Minitab, SYSTAT and SAS input data formats are supported by SigmaPlot.

Axon binary and text electrophysiology files may be imported. Also the Electrophysiology module, purchased separately, allows importing specific parts of electrophysiology files from Axon Instruments ABF files, Bruxton Corporation’s Acquire format and HEKA electronik’s Pulse format. Import any ODBC compliant database. Excel and Access database files are supported. Run SQL queries on tables and selectively import information.

#### SigmaPlot Features

##### Choose from a wide range of graph types to best present your results

SigmaPlot provides more than 100 different 2-D and 3-D graph types. From simple 2-D scatter plots to compelling contour, Forest and radar plots, SigmaPlot gives you the exact technical graph type you need for your demanding research. With so many options, you can always find the best visual representation of your data.

##### Statistical Analysis is no longer a daunting task

SigmaPlot now offers almost 50 of the most frequently used statistical tests in scientific research by integrating SigmaStat into one application. Suggestion of the most appropriate statistical tests is offered by a software-based Advisor. Raw and indexed data formats are accepted to avoid data reformatting.

Violation of data assumptions is checked in the background and, if true, the correct test to use is recommended. Reports with descriptive interpretations are generated and graphs specific to each test may be created.

SigmaPlot now employs an all new user interface allowing users to easily setup a global curve fit. This gives users the ability to easily share one or more equation parameters across multiple data sets.

Non-linear curve fitting is known to produce incorrect results in some instances.The problem is that you don´t necessarily know that this has happened. Dynamic Curve Fitting is designed to determine if this has happened and if so what the likely best fit is.

Export your graphs as high-resolution, dynamic Web pages – not simple GIF or JPEG files. Viewers can explore data used to create graphs and zoom, pan or print images at full resolution directly from a Web Browser. Automatically generate active Web objects from your graphs or embed the objects within other Web pages.

- Simply select the Web graph to share its data with colleagues and students
- Share the data behind your graphs with colleagues and students
- Enable colleagues to print your full report from your intranet or Web site directly from their browsers – without compromising the quality of the graphs
- Create an optional password while exporting your graph to limit data access to authorized users
- Produce Web documents without knowing HTML or embed SigmaPlot Web object graphs within HTML files to create interactive electronic reports

Each worksheet can hold a list of user defined transforms that will automatically be re-run whenever the transform input data has changed.

Let’s say you would like to start by selecting a particular kind of graph but you don’t know how to set up the worksheet to achieve it. SigmaPlot lets the user select a graph first and then gives you a pre-formatted worksheet to structure their data. The data entered into the worksheet is immediately displayed on the graph. This feature can demonstrate to you the strong relationship between the data format and the graph type.

##### Import

- Excel, ASCII Plain Text, Comma Delimited, MS Access
- General ASCII import filter
- SigmaPlot DOS 4.0, 4.1, 5.0 data worksheets, SigmaPlot 1.0, 2.0 Worksheet, and 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0 and 11.0 Windows, SigmaPlot 4.1 and 5.0 Macintosh data worksheets
- Comma delimited and general purpose ASCII import filter
- Symphony, Quattro Pro, dBASE E, DIF, Lotus 1-2-3, Paradox
- SigmaStat DOS and 1.0 worksheets, SYSTAT, SPSS, SAS data set V6. V8, V9, SAS export file, Minitab V8 to V12
- SigmaScan, SigmaScan Pro, SigmaScan Image, Mocha
- TableCurve 2D and 3D
- Axon Binary, Axon Text
- Import ODBC compliant databases
- Run SQL queries on tables and selectively import information
- Import Excel 2007 files directly into SigmaPlot

##### Export

- Excel, ASCII Plain Text, Tab delimited, Comma delimited
- SigmaPlot 1.0, 2.0, and 3.0 for Windows, SigmaPlot 5.0 for Macintosh data worksheets
- SigmaPlot 7.101, 8.0, 9.0, 10.0, 11.0
- SigmaScan, SigmaScan Pro, SigmaScan Image, Mocha
- SigmaStat 2.0, SYSTAT, SAS V6 data set, Minitab V11
- Vector PDF and HTML export of graphs and reports
- Symphony, Quattro Pro, dBASE III, DIF, Lotus 1-2-3, Paradox
- Graph formats: JPEG, GIF, PNG, HTML, TIFF CMYK, TIFF RGB, Bitmap, Metafile (wmf), Enhanced Metafile (emf), PDF, PSD, EPS, PDF vector, SVG, SWF

#### Graphing software that makes data visualization easy

Graph creation starts with SigmaPlot’s award-winning interface. Take advantage of ribbon collections of common properties, tabbed selection of graphs, worksheets and reports, right mouse button support and graph preferences. Select the graph type you want to create from the Graph Toolbar’s easy-to-read icons. The interactive Graph Wizard leads you through every step of graph creation. You get compelling, publication-quality charts and graphs in no time. SigmaPlot offers more options for charting, modeling and graphing your technical data than any other graphics software package.

Compare and contrast trends in your data by creating multiple axes per graph, multiple graphs per page and multiple pages per worksheet. Accurately arrange multiple graphs on a page using built-in templates or your own page layouts with SigmaPlot’s WYSIWYG page layout and zoom features.

#### Customize every detail of your charts and graphs

SigmaPlot offers the flexibility to customize every detail of your graph. You can add axis breaks, standard or asymmetric error bars and symbols; change colors, fonts, line thickness and more. Double-click on any graph element to launch the Graph Properties dialog box. Modify your graph, chart or diagram further by pasting an equation, symbol, map, picture, illustration or other image into your presentation. And select anti-aliasing to display jaggy-free smooth lines that can be used in your PowerPoint^{®} presentations.

#### Quickly Plot your Data from Existing Graph Templates in the Graph Style Gallery

Save all of the attributes of your favorite graph style in the Graph Style Gallery. Add greater speed and efficiency to your analysis by quickly recalling an existing graph type you need and applying its style to your current dataset.

- Quickly save any graph with all graph properties as a style and add a bitmap image to the gallery
- No need to be an expert, create customized graphs in no time with the Graph Gallery
- Choose an image from the Graph Style Gallery to quickly plot your data using an existing graph template
- Save time by using a predetermined style to create a graph of the data
- Avoid re-creating complex graphs

But, remember, you don’t necessarily need to use the power of the Graph Gallery since every graph in SigmaPlot is a template. In the Notebook Manager, you can copy and paste a graph from one worksheet to another and all the attributes of that graph are applied to the new data saving much time.

#### Publish your charts and graphs anywhere

Create stunning slides, display your graphs in reports or further customize your graphs in drawing packages. Save graphs for publication in a technical journal, article or paper with SigmaPlot’s wide range of graphic export options. Presenting and publishing your results has never been easier – or looked this good.

Create customized reports with SigmaPlot’s Report Editor or embed your graphs in any OLE (Object Linking and Embedding) container – word processors, Microsoft PowerPoint or another graphics program. Then, just double click your graph to edit directly inside your document. Quickly send your high-resolution graphs online to share with others.

Export your graphs ashigh-resolution, dynamic Web pages – not simple GIF or JPEG files. Viewers can explore data used to create vector graphs and zoom, pan or print images at any resolution directly from a Web Browser. Automatically generate active Web objects from your graphs or embed the objects within other Web pages.

- Share the data behind your web-based graphs with colleagues and students
- Enable colleagues to print your full report from your intranetor Web site directly from their browsers – without compromising the quality of the graphs
- Create an optional password while exporting your graph to limit data access to authorized users
- Produce Web documents without knowing HTML, or embed SigmaPlot Web object graphs in existing HTML files to create interactive electronic reports

#### Data Analysis Doesn’t Get Much Easier

SigmaPlot provides all the fundamental tools you need to analyze your data from basic statistics to advanced mathematical calculations. Click the View Column Statistics button to instantly generate summary statistics including 95% and 99% confidence intervals. Run t-tests, linear regressions, non-linear regressions and ANOVA with ease. You can fit a curve or plot a function and get a report of the results in seconds. Use built-in transforms to massage your data and create a unique chart, diagram or figure. With SigmaPlot – it’s all so simple!

#### Use SigmaPlot within Microsoft Excel

Access SigmaPlot right from your active Microsoft Excel worksheet. Tedious cut-and-paste data preparation steps are eliminated when you launch SigmaPlot’s Graph Wizard right from the Excel toolbar. Use Excel in-cell formulas, pivot tables, macros and date or time formats without worry. Keep your data and graphs in one convenient file.

#### Transforms and Quick Transforms

Generate simulated data or modify worksheet columns of data with transforms. Create simple one-line transforms with the Quick Transforms feature that walks you through transform implementation. Or create extremely complex transforms with hundreds of lines of code.

#### Use the Regression Wizard to fit data easily and accurately

Fitting your data is easy with the SigmaPlot Regression Wizard. The Regression Wizard automatically determines your initial parameters, writes a statistical report, saves your equation to your SigmaPlot Notebook, and adds your results to existing graphs or creates a new one!

The Regression Wizard accurately fits nearly any equation – piecewise continuous, multifunctional, weighted, Boolean functions and more – up to 10 variables and 25 parameters. You can even add your own curve fit equations and add them to the Regression Wizard.

#### Use the Dynamic Curve Fitter to determine if your fit is valid

The Dynamic Curve Fitter performs 200 or more curve fits using your equation and data starting from optimally different initial starting values. The results are ranked by goodness of fit so that you can check the top ranked results against the result you obtained from the Regression Wizard.

For many simple equations, which are fit to data sets with a sufficiently large number of data points, the Dynamic Curve Fitter finds the same result as the Regression Wizard. But the problem is that the user simply does not know whether the solution found by the Regression Wizard is the best possible or not. So there is always a concern that the correct solution has not been found. Dynamic fitting minimizes this concern. Its use is encouraged prior to publishing results particularly if a complicated equation is used.

#### Plot Nearly ANY Mathematical Function

Plotting user-defined and parameterized equations is only a mouseclick away with the Plot Equation feature. Just type the function or select one from the built-in library and specify the parameters and the range. It’s that simple! Create your own built-in functions and save them for future use. Plot functions on new or existing graphs or plot multiple functions simultaneously using different parameter values. Save plotted X and Y results to the worksheet.

#### Maximize your Productivity with SigmaPlot’s Automation

#### Automate Complex Repetitive Tasks

Create macros in no time with SigmaPlot’s easy-to-use macro language. Not a programmer? No problem. With SigmaPlot, you can record macros by point-and-click with the macro recorder. Use macros to acquire your data, execute powerful analytical methods, and create industry-specific or field-specific graphs. Use one of the thirty built-in macros as provided or use these macros as a base to quickly create your own macros.

Share the power of SigmaPlot with less-experienced users by using macros to tailor the SigmaPlot interface for your particular application. Create custom dialog boxes, menu choices and forms to help guide novice users through a session.

**Tap into SigmaPlot’s Powerful Capabilities from Within Other Applications**

Call on SigmaPlot´s functionality from external sources that have Visual Basic embedded including Microsoft Word^{®}, Microsoft Excel^{®}, Microsoft PowerPoint^{®} or custom software applications. Analyze and graph your data using SigmaPlot within those applications.

For example, you can run a Visual Basic script in Microsoft Word^{®} or Excel^{®} that calls on SigmaPlot to generate and embed your graph in the document. SigmaPlot´s OLE2 automation provides unlimited flexibility.

##### SigmaPlot Has Complete Advisory Statistical Analysis Features

SigmaPlot is now a complete graphing AND an advisory statistics suite. All of the advanced statistical analysis features found in the package known as SigmaStat have now been incorporated into SigmaPlot along with several new statistical features. SigmaPlot guides users through every step of the analysis and performs powerful statistical analysis without the user being a statistical expert.

In addition to the EC50 value already computed, the user can also compute other user-entered EC values such as EC40 and EC60 and compute them instantly. Two five-parameter logistic functions have also been added and the Dynamic Curve Fitting feature included to help solve difficult curve fitting problems.

In earlier versions of SigmaPlot, almost all objects in a 2D graph were selectable with just a mouse click. However, almost all objects in a 3D graph were not. SigmaPlot now adds mouse selectability of all 3D graph objects with the ability to customize all 3D objects.

##### New Worksheet Features Include

- Import Excel worksheet data into a SigmaPlot worksheet or Open an Excel worksheet as an Excel worksheet in SigmaPlot
- Mini toolbar for worksheet cell editing
- Zoom enabled worksheet
- Worksheet scrolling with mouse wheel
- Line widths may be placed in the worksheet for graph customization
- Formatted text (subscript, etc.) in worksheet cells

Create stunning slides, display your graphs in reports or further customize your graphs in drawing packages. Save graphs for publication in a technical journal, article or paper with SigmaPlot’s wide range of graphic export options. Presenting and publishing your results has never been easier – or looked this good. Create customized reports with SigmaPlot’s Report Editor or embed your graphs in any OLE container – word processors, Microsoft PowerPoint or graphics program. Just double click your graph to edit directly inside your document. Quickly send your high-resolution graphs online to share with others.

##### SigmaPlot’s Notebook Functionality

- Can hold SigmaPlot worksheets, Excel worksheets, reports, documents, regression wizard equations, graph pages, and macros.
- New dialog-bar-based notebook that has several states: docks, re-sizable, hide-able, summary information mode, etc.
- Browser-like notebook functionality that supports drag-n-drop capabilities
- Direct-editing of notebook summary information

##### Automate Routine and Complex Tasks

- Visual Basic compatible programming using built-in macro language interface
- Macro recorder to save and play-back operations
- Full automation object support – use Visual Basic to create your own SigmaPlot-based applications
- Run built-in macros or create and add your own scripts
- Add menu commands and create dialog boxes
- Toolbox ribbon: helpful macros appear as separate grouped items
- Export graph to PowerPoint Slide (macro)
- “Insert Graph to Microsoft Word” Toolbox ribbon macro
- Keyboard shortcuts in the Graph page and most Microsoft Excel keyboard shortcuts in the worksheet.

##### Symbol Types

- Over 100 symbol types
- 30 new symbol types that include half-filled and BMW styles
- Edit font when using text as symbol
- Access new symbols directly from graph properties dialog, toolbar, legend page and the symbol dialog box
- More line types such as dash and gap patterns
- More fill patterns for bar charts and area plots, that can be independently set from the line color

##### SigmaPlot Report Editor

- Cut and paste or use OLE to combine all the important aspects of your analysis into one document.
- Copy / Paste tabular data between report and Excel worksheet
- Choose from a wide range of styles, sizes and colors from any system font
- New tables with pre-defined styles or user customized
- Export to most word processors
- Add decimal tabs, tab leader, true date/time fields
- Vertical and horizontal rulers for report formatting
- Auto-numbering
- Change report background color
- Improved formatting ruler
- Zoom enabled in reports
- Drag and drop Word 2007 and 2010 documents into reports

##### Page Layout and Annotation Options

- OLE 2 container and server
- Automatic or manual legends
- True WYSIWYG
- Multi line text editor
- Multiple curves and plots on one graph
- Multiple axes on one graph
- Arrange graphs with built-in templates
- Multiple levels of zooming and custom zooming
- Scale graph to any size
- Resize graphic elements proportionally with resizing graph
- Alignment and position tools
- Draw lines, ellipses, boxes, arrows
- Layering options
- Over 16 million custom colors
- Inset graphs inside one another
- Selection of graph objects
- Right-click property editing
- New zoom, drag and pan controls
- Mouse wheel scrolling enabled
- Right-click property editing for 3D Graphs
- Color schemes
- Paste graphic objects from other

## General Features

#### SigmaPlot Helps You Quickly Create Exact Graphs

With the new Graph Properties user interface you can select the property category in the tree on the left and then change properties on the right. The change is immediately graphed and if you move your cursor off the panel then it becomes transparent and you can see the effect of your changes without leaving the panel.

The “select left and change right” procedure makes editing your graphs quick and easy. SigmaPlot takes you beyond simple spreadsheets to help you show off your work clearly and precisely. With SigmaPlot, you can produce high-quality graphs without spending hours in front of a computer. SigmaPlot offers seamless Microsoft Office^{®} integration, so you can easily access data from Microsoft Excel^{®} spreadsheets and present your results in Microsoft PowerPoint^{®} presentations.

The user interface also includes Microsoft Office style ribbon controls. And the tabbed window interface efficiently organizes your worksheets and graphs for easy selection. And these tabs may be organized into either vertical or horizontal tab groups. Graph Gallery and Notebook Manger panes may be moved to any position and easily placed using docking panel guides. You can add frequently used objects to the Quick Access Toolbar. For example you might want to add Notebook Save, Close All, Refresh Graph Page and Modify Plot.

## New Features

#### New Features and Improvements in SigmaPlot Version 14.5

- Forest Plots
- Kernel Density Plots
- 10 New Color Schemes
- Dot Density Graph with mean and standard error bars
- Legend Improvements
- Horizontal, Vertical and Rectangular Legend Shapes
- Cursor over side or upper or lower handle

- User interface to set number of legend item columns in the Properties dialog. The permissible column numbers are displayed in the combo list
- Change the number of legend item columns by selecting and dragging the middle handle in the bounding box
- Reorder legend items
- Through properties dialog – move one or multiple legend items up or down using the up/down control on top of the list box
- Through cursor movement – move one or multiple legend items up or down. Select the legend item(s) and use keyboard up and down arrow key for movement within the bounding box
- Through mouse select and cursor movement for items in the bounding box

- Individual legend items property settings – select individual legend items and use the mini tool bar to change the properties
- Legend box blank region control through cursor
- Cursor over corner handle
- allows proportional resizing

- Add simple direct labeling
- Support “Direct Labeling” in properties dialog using the checkbox control “Direct Labeling”
- Ungroup legend items – the individual legend items can be moved to preferred locations and move in conjunction with the graph

- Legend Title support has been added (no title by default). The user can add a title to the legend box using the legend properties panel
- Reverse the legend items using the right click context menu
- Open Legend Properties by double clicking either Legend Solid or Legend Text
- Reset has been added to legends to reset legend options to default

- Horizontal, Vertical and Rectangular Legend Shapes

New Analysis Features

- Principal Component Analysis (PCA)
- Analysis of Covariance (ANCOVA)
- Added P values to multiple comparisons for non-parametric ANOVAs
- Removed the combo box choices for multiple comparison significant levels and tied the significance level of multiple comparisons to the main (omnibus) test
- Added the Akaike Information Criterion to Regression Wizard and Dynamic Fit Wizard reports and the Report Options dialog
- Added back the Rerun button in the SigmaStat group
- Updated the fit library standard.jfl
- Added probability functions, to now include 24, for curve fitting or function visualization
- The tolerance value for all equations has been modified to use “e-notation” instead of fixed decimal. This allows the user to read the value without scrolling.
- Add seven weighting functions to all curve fit equations in standard.jfl. There is a slight variant added for 3D equations.

#### New User Interface Features

- Rearrange Notebook items in a section by dragging
- New SigmaPlot tutorial PDF file
- Line widths from a worksheet column

#### New Import/Export Features

- Added the SVG and SWF file formats for scalable vector graphics export
- Added Vector PDF export to improve on the existing raster PDF
- File import and export support is added for Versions 13 and 14 of Minitab, Version 9 of SAS, Version 19 of SPSS and Version 13 of Symphony

#### SigmaPlot Product Features

**Forest Plot**

A forest plot is one form of “meta-analysis” which is used to combine multiple analyses addressing the same question. Meta-analysis statistically combines the samples of each contributing study to create an overall summary statistic that is more precise than the effect size in the individual studies. Individual study values and their 95% confidence intervals are shown as square symbols with horizontal error bars and the overall summary statistic as a diamond with width equal to its 95% confidence interval.

**Kernel Density**

The kernel density feature will generate an estimate of the underlying data distribution. This should be compared to the step-like histogram. It has advantages (no bars) and disadvantages (loss of count information) over a histogram and should be used in conjunction with the histogram. They can be created simultaneously.

**Dot Density with Mean & Standard Error Bars**

The mean plus standard error bar computation, symbol plus error bars, has been added to the Dot Density graph. This enhances the other possible dot density display statistics – mean, median, percentiles and boxplot.

**New Color Schemes**

Ten new color schemes have been implemented. Three examples are shown below:

**Legend Improvements – ****Shapes**

Vertical, horizontal and rectangular legend shapes are now available.

**Reverse Legend Order**

You can now select to reverse the legend item order. This provides a more logical order for some graph types.

**Reorder Legend Items**

There are three ways to reorder the legend items. As shown here, you can move one or multiple legend items up or down using the up/down arrow controls in the Legends panel of Graph Properties. Even easier, just select the item in the legend and use the keyboard up and down arrow keys. Or select the legend item and drag it to the new position with the mouse cursor.

**Mini-Toolbar Editing of Legend Items**

Legend items may now be edited by clicking on the item and using the mini-toolbar.

**Direct Labeling**

The legend can now be ungrouped and individual legend items placed adjacent to the appropriate plots. The labels will move with the graph to maintain position with respect to the graph. Since the label is adjacent to the plot, visual identification of each plot is now much easier.

**Principal Component Analysis (PCA)**

Principal component analysis (PCA) is a technique for reducing the complexity of high-dimensional data by approximating the data with fewer dimensions. Each new dimension is called a principal component and represents a linear combination of the original variables. The first principal component accounts for as much variation in the data as possible. Each subsequent principal component accounts for as much of the remaining variation as possible and is orthogonal to all of the previous principal components.

You can examine principal components to understand the sources of variation in your data. You can also use them in forming predictive models. If most of the variation in your data exists in a low-dimensional subset, you might be able to model your response variable in terms of the principal components. You can use principal components to reduce the number of variables in regression, clustering, and other statistical techniques.

The primary goal of Principal Components Analysis is to explain the sources of variability in the data and to represent the data with fewer variables while preserving most of the total variance.

Graphical output consists of Scree, Component Loadings and Component Scores plots.

**Analysis of Covariance (ANCOVA)**

A single-factor ANOVA model is based on a *completely randomized design* in which the subjects of a study are randomly sampled from a population and then each subject is randomly assigned to one of several factor levels or treatments so that each subject has an equal probability of receiving a treatment. A common assumption of this design is that the subjects are *homogeneous*. This means that any other variable, where differences between the subjects exist, does not significantly alter the treatment effect and need not be included in the model. However, there are often variables, outside the investigator’s control, that affect the observations within one or more factor groups, leading to necessary adjustments in the group means, their errors, the sources of variability, and the P-values of the group effect, including multiple comparisons.

These variables are called *covariates*. They are typically continuous variables, but can also be categorical. Since they are usually of secondary importance to the study and, as mentioned above, not controllable by the investigator, they do not represent additional main-effects factors, but can still be included into the model to improve the precision of the results. Covariates are also known as *nuisance *variables or *concomitant* variables.

ANCOVA (Analysis of Covariance) is an extension of ANOVA obtained by specifying one or more covariates as additional variables in the model. If you arrange ANCOVA data in a SigmaPlot worksheet using the *indexed* data format, one column will represent the factor and one column will represent the dependent variable (the observations) as in an ANOVA design. In addition, you will have one column for each covariate. When using a model that includes the effects of covariates, there is more explained variability in the value of the dependent variable.

This generally reduces the unexplained variance that is attributed to random sampling variability, which increases the sensitivity of the ANCOVA as compared to the same model without covariates (the ANOVA model). Higher test sensitivity means that smaller mean differences between treatments will become significant as compared to a standard ANOVA model, thereby increasing statistical power.

As a simple example of using ANCOVA, consider an experiment where students are randomly assigned to one of three types of teaching methods and their achievement scores are measured. The goal is to measure the effect of the different methods and determine if one method achieves a significantly higher average score than the others. The methods are Lecture, Self-paced, and Cooperative Learning. Performing a One Way ANOVA on this hypothetical data gives the results in the table below, under the ANOVA column heading. We conclude there is no significant difference among the teaching methods. Also note that the variance unexplained by the ANOVA model which is due to the random sampling variability in the observations is estimated as 35.17.

It is possible that students in our study may benefit more from one method than the others, based on their previous academic performance. Suppose we refine the study to include a covariate that measures some prior ability, such as a state-sanctioned Standards Based Assessment (SBA). Performing a One Way ANCOVA on this data gives the results in the table below, under the ANCOVA column heading.

ANOVA | ANCOVA | |||

Method | Mean | Std. Error | Adjusted Mean | Std. Error |

Coop | 79.33 | 2.421 | 82.09 | 0.782 |

Self | 83.33 | 2.421 | 82.44 | 0.751 |

Lecture | 86.83 | 2.421 | 84.97 | 0.764 |

P = 0.124 | P = 0.039 | |||

MSres = 35.17 | MSres = 3.355 |

The adjusted mean that is given in the table for each method is a correction to the group mean to control for the effects of the covariate. The results show the adjusted means are significantly different with the Lecture method as the more successful. Notice how the standard errors of the means have decreased by almost a factor of three while the variance due to random sample variability has decreased by a factor of ten. A reduction in error is the usual consequence of introducing covariates and performing an ANCOVA analysis.

There are four ANCOVA result graphs – Regression Lines in Groups, Scatter Plot of Residuals, Adjusted Means with Confidence Intervals, and Normality Probability Plot:

**P Values for Nonparametric ANOVAs**

The non-parametric ANOVA tests in SigmaPlot are the Kruskal-Wallis test (One-Way ANOVA on Ranks) and the Friedman test (One-Way Repeated Measures ANOVA on Ranks). Both of these provide four post-hoc testing procedures to determine the source of significant effects in the treatment factor. The four procedures are Tukey, SNK, Dunn’s, and Dunnett’s.

The first three procedures can be used to test the significance of each pairwise comparison of the treatment groups, while the last two can be used to test the significance of comparisons against a control group. Dunn’s method is the only procedure available if the treatment groups have unequal sample sizes.

When a post-hoc testing procedure is used, a table is given in the report listing the results for the pairwise comparisons of the treatment levels. The last column of the table shows whether the difference in ranks is significant or not. In previous versions of SigmaPlot there is no adjusted p-value given that can be compared to the significance level of the ANOVA (usually .05) to determine significance.

This is because SigmaPlot had been determining significance by comparing the observed test statistic, computed for each comparison, to a critical value of the distribution of the statistic that is obtained from a lookup table. SigmaPlot had two sets of lookup tables for the probability distributions corresponding to the four post-hoc methods, where one set was for a significance level of .05 and another set was for a significance level of .01.

This was recently changed to use analytical procedures to compute the p-values of these distributions, making the lookup tables obsolete. Because of this change, we are now able to report the adjusted p-values for each pairwise comparison. This change also makes it possible to remove the restriction of using .05 and .01 as the only significance levels for multiple comparisons. Thus the user can enter any valid P value significance level from 0 to 1.

**Akaike Information Criterion (AICc)**

The *Akaike Information Criterion (AIC) *provides a method for measuring the relative performance in fitting a regression model to a given set of data. Founded on the concept of *information entropy*, the criterion offers a relative measure of the information lost in using a model to describe the data. More specifically, it gives a tradeoff between maximizing the likelihood for the estimated model (the same as minimizing the residual sum of squares if the data is normally distributed) and keeping the number of free parameters in the model to a minimum, reducing its complexity. Although goodness-of-fit is almost always improved by adding more parameters, *overfitting* will increase the sensitivity of the model to changes in the input data and can ruin its predictive capability.

The basic reason for using AIC is as a guide to model selection. In practice, it is computed for a set of candidate models and a given data set. The model with the smallest AIC value is selected as the model in the set which best represents the “true” model, or the model that minimizes the information loss, which is what AIC is designed to estimate. After the model with the minimum AIC has been determined, a *relative likelihood *can also be computed for each of the other candidate models to measure the probability of reducing the information loss relative to the model with the minimum AIC. The relative likelihood can assist the investigator in deciding whether more than one model in the set should be kept for further consideration.

The computation of AIC is based on the following general formula obtained by Akaike

whereis the number of estimable parameters in the regression problem, which includes the model parameters and the unknown variance of the observations, and is the maximized value of the likelihood function for the estimated model.

When the sample size of the data is small relative to the number of parameters (some authors say when is not more than a few times larger than), AIC will not perform as well to protect against overfitting. In this case, there is a corrected version of AIC given by

It is seen that AICc imposes a greater penalty than AIC when there are extra parameters. Most authors seem to agree that AICc should be used instead of AIC in all situations and it is AICc that is implemented in SigmaPlot. The Asymmetric equation in the graph is significantly better than the Symmetric since its AICc value is greater than 7 units less than the Symmetric equation – a rule of thumb for AICc. If the difference is greater than 2 then the equation with the smaller AICc value should not be considered as the best but rather a candidate for the best equation.

**Nonlinear Regression Probability Functions**

24 new probability fit functions have been added to the fit library standard.jfl. These functions and some equations and graph shapes are shown below.

**Nonlinear Regression Weighting Functions**

There are now seven different weighting functions built into each nonlinear regression equation (3D are slightly different). These functions are reciprocal y, reciprocal y squared, reciprocal x, reciprocal x squared, reciprocal predicteds, reciprocal predicteds squared and Cauchy. The iteratively reweighted least squares algorithm is used to allow the weights to change during each nonlinear regression iteration. In this way “weighting by predicteds”, a commonly used method, can be obtained by selecting the reciprocal_pred weighting option.

Also, Cauchy weighting (select weight_Cauchy) can be used to fit an equation to data that contains outliers and the effect of the outliers will be minimized. Users can create their own weighting methods in terms of residuals and/or parameters to implement other robust fitting methods. The equation section of a fit file is shown with the seven built-in weighting functions.

**User Interface Features – ****Rearrange items in your notebook by dragging**

Objects in a notebook section are not necessarily created in a logical order. You can now drag items within a section to new positions to place them more logically.

**An Updated SigmaPlot Tutorial**

The new tutorial makes creating graphs for the first time easy. It starts with simple examples and gradually becomes more complex.

**Specify Plot Line Widths from a Worksheet Column**

Line width values can now be entered in a worksheet column. These values may be used within a graph or across multiple graphs on the page.

**New Vector Export File Formats**

SVG (Scalable Vector Graphics), SWF (Adobe Flash Player) and Vector PDF file formats have been added. These are scalable formats where no resolution is lost when zooming to different levels. SVG is the standard graphics format for the web and SWF can be used with Adobe Flash Player. Because pdf is used so frequently, the vector PDF format is now attached to the Create PDF button on the Home ribbon.

**Updated Application File Formats**

File import and export support has been updated to Versions 13 and 14 of Minitab, Version 9 of SAS and Version 19 of SPSS.

## System Requirements

#### Hardware

- 2 GHz 32-bit (x86) or 64-bit (x64) Processor
- 2 GB of System Memory for 32-bit (x86)
- 4 GB of System Memory for 64-bit (x64)
- 300 MB of Available Hard Disk Space
- CD-ROM Drive or Internet Connection
- 800×600 SVGA/256 Color Display or better
- Internet Explorer Version 8 or better

#### Software

- Windows XP, Windows Vista, Windows 7, Windows 8.x, Windows 10; Internet Explorer 6 or higher
- Office 2003 or higher (paste to Powerpoint Slide, Insert Graphs into Word and other macros)