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# Manual InnerSoft STATS

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InnerSoft STATS is a Descriptive Statistics Application. InnerSoft STATS compute statistics for parameter estimation and Statistical hypothesis testing. Descriptive Statistics: Mean, Variance, Standard deviation, Coefficient of variation, Quartiles, Percentiles, Skewness, Kurtosis, Mode, Interquartile range, Sum of Squares. One-Sample Test: One-sample z-test, One-sample t-test, Chi-squared test for variance.Two-Sample Test: Student's t-test for Independent samples (pooled t-test for equal variances and unpooled t-test for unequal variances), Student's t-test for Paired samples, Two-sample F-test of equality of variances.One-Way ANOVA with multiple comparisons methods: Scheffe, Tukey HSD, Sidak, Fisher LSD, Bonferroni. Welch’s Test for equality of means, Brown–Forsythe Test for equality of means. Homoscedasticity Test: Levene's Test, Brown–Forsythe Test for equality of variances, Bartlett's Test

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### Manual InnerSoft STATS

1. 1. InnerSoft STATS Application Manual v0.4
2. 2. APPLICATION MANUAL V0.4 InnerSoft STATS 2 INDEX 1. Installation and Activation 1.1. Installation 1.2. Trial Version 1.3. Activation 1.4. License 1.4.1. Upgrades 1.4.2. License Renewal 2. Introduction to ISSTATS 2.1. Open and Save files 2.2. Technical Limitations and Recommendations 3. Working in the Main window 3.1. File Menu 3.2. Analyze Menu 3.3. Help Menu 4. Working in the Worksheet window 4.1. File Menu 4.2. Edit Menu 4.3. Options Menu 5. Working in the Output window 6. Descriptive Statistics 7. One Sample Test 7.1. Z Test 7.2. T Test 7.3. Variance Test 8. Two-Sample Test 8.1. 2-Sample t-Test 8.2. Paired t-Test 8.3. 2 Variances F-Test 9. One-Way ANOVA 10. Homoscedasticity Tests 11. Methods and Formulas Help
3. 3. APPLICATION MANUAL V0.4 InnerSoft STATS 3 1. – Installation and Activation 1.1. – Installation Click on setup.exe to install the application. 1.2. – Trial Version InnerSoft STATS works initially as limited trial version. You will not be allowed to use following tools:  Open or Save any Project or Worksheet.  Using Paste or Insert when editing cells. 1.3. – Activation In order to use InnerSoft STATS Full Version, user must purchase a license before activating the product. To purchase InnerSoft STATS Full Version, you must first contact with us by email innersoft@itspanish.org innersoft@gmail.com You can also call us (Spain number phone): There are different payment methods that you can find at: http://isstats.itspanish.org/index.htm#buy Once product has been paid, you can activate it. Just enter Help/Activate ISSTATS and follow instructions. For a successful activation process, InnerSoft requires permissions to read/write files in its installation folder, usually: 'C:Program FilesISSTATS'. So may you have to login to Windows as Administrator or have to modify User Account Control in order to activate successfully InnerSoft STATS.
6. 6. APPLICATION MANUAL V0.4 InnerSoft STATS 6 XML  XML format is human readable and editable in plain text  Can be imported by lot of applications (Excel…)  Takes high size: storing a variable with 1’5 millions of values will take around 50 Mb in a XML file  Has a high compression rate: a XML file of 50 Mb will get only 1 Mb of size once compressed in zip format  Opening and saving is fast Binary  Binary format is not human readable  Can be only open by ISSTATS  Takes less size than XML: storing a variable with 1’5 millions of values will take around 30 Mb in a Binary file.  Has a low compression rate: a Binary file of 50 Mb will get 6 Mb of size once compressed in zip format.  Opening and saving is slow. Thus XML compressed in zip is the better way of storing large amounts of data. 2.2. – Technical Limitations and Recommendations ISSTATS uses decimal data type instead of double data type. Decimal is specially used for financial and monetary calculation which requires higher accuracy. It is not adequate for scientific applications because of its lower range: the largest value is +/–7.9228162514264337593543950335, and the smallest nonzero value is +/–0.0000000000000000000000000001 (+/–1E–28). So if you need to analyses social or scientific data with large numbers (astronomy distances, human stupidity…), just use a different statistical package. ISSTATS work with on–memory data (no database storage). Thus it can compute quickly. In the other hand, memory usage is huge. ISSTATS can handle millions of data. It can compute all descriptive parameters of a variable with 1’5 million values in only a few seconds (Processor: Pentium R Dual Core CPU E 6600 @ 3.06 GHz 3.07 GHz RAM: 8.0 GB SO: Windows 7 64 Bits). In the other hand, the Data table can easily crash if you try to manipulate this huge data. Operations like copying, pasting, removing and selecting cells in such a big table can hang the application. It really depends on the total number of cells in the table. To avoid this problem, limit the number of table columns when operating with huge data. If you want to handle variables with millions of values, set the number of table columns to minimum in the Worksheet OPTIONS menu: if you are using only 4 variables, set only 4 columns. You should perform operations with columns as much as possible. If you want to clear a variable, select the column and clear the column; avoid selecting all cells in the column and clearing these cells. If you want to copy a variable then copy the column, not the cells. Actually, undo/redo history has no limits. A limit will be implemented in a future. If you feel that ISSTATS is consuming too much memory in your PC, you should save your project, restart the application and open your project again. This will clear the undo/redo history. Please, remember this is 0.4 version.
7. 7. APPLICATION MANUAL V0.4 InnerSoft STATS 7 3. – Working in the Main window 3.1. – File Menu New Project Creating a new project closes the current project. If your current project has been changed since it was last saved, ISSTATS will prompt you to save all or part of the project before closing it. Open Project Opening a new project closes the current project. If your current project has been changed since it was last saved, ISSTATS will prompt you to save all or part of the project before closing it. Save Project Saves the current project in a XML or Binary file. If the project has been saved before, Save Project saves the file with the current file name without displaying a dialog box. If the project has not been saved before, Save Project opens the Save Project As dialog box. When you save the project, you save the input information about your work:  the contents of project settings and project information  the columns of data in the Worksheet window The output document will not be saved in your project file. Save Project As Saves the current project in a XML or Binary file with a different name. If the project has not been saved before, choosing File > Save Project also opens this dialog box. Dialog box items  Save in: Choose a drive and folder.  File name: Enter a file name.  Save as type: Choose a file type from the list. o XML files (*.xml) o DAT files (*.dat) Exit Exits you from ISSTATS. ISSTATS will prompt you to save the changes before closing the application. 3.2. – Analyze Menu Descriptive Statistics: See Chapter 6. – Descriptive Statistics One Sample Test: See Chapter 7. – One Sample Test Two-Sample Test: See Chapter 8. – Two-Sample Test
8. 8. APPLICATION MANUAL V0.4 InnerSoft STATS 8 One-Way ANOVA: See Chapter 9. – One-Way ANOVA Homoscedasticity Tests: See Chapter 10. – Homoscedasticity Tests 3.3. – Help Menu Contents Open this PDF. Buy ISSTATS Open a web page with information about buying InnerSoft STATS. Activate ISSTATS Open the menu to activate ISSTATS in order to use all features. About Shows ISSTATS version and contact information.
9. 9. APPLICATION MANUAL V0.4 InnerSoft STATS 9 4. – Working in the Worksheet window 4.1. – File Menu New Worksheet Creating a new worksheet adds an empty worksheet to the current project. An empty Data window will appear. Open Worksheet Copies data from a file into the current project. When you open a file, you copy the contents of the file into the current project. Any changes you make to the file while in the project will not affect the original file. Save Worksheet Saves the current worksheet in the same file and format as the file name displayed in the window title. If the current worksheet is untitled, File > Save Worksheet brings up a dialog box where you can specify a file name and location in which to save your data. See File > Save Worksheet As for a description of this dialog box. Choose File > Save Worksheet As to rename your worksheet or save it to a new location. Save Worksheet As Saves the worksheet data in a file. Use this command if you want to rename your worksheet or save it to a new location. Dialog box items  Save in: Choose a drive and folder.  File name: Enter a file name.  Save as type: Choose a file type from the list. o XML files (*.xml) o DAT files (*.dat) Export to Excel This tool will export to Excel only those columns that has a variable defined. Empty columns will not be exported. Exporting to Excel has some limitations. As worksheet limit size in Excel 2010 is 65,536 rows, ISSTATS will divide the data collection in different sheets of 65530 rows everyone. If you are going to export 1’5 million values, the application will create 23 different Excel worksheets with 65530 rows each one. As columns limit in Excel 2010 is 256 columns, ISSTATS will not export the data of more than 256 variables. 4.2. – Edit Menu Undo & Redo
10. 10. APPLICATION MANUAL V0.4 InnerSoft STATS 10 To undo an action, click Undo Button on the Toolbar. To redo an action that you undid, click Redo Button on the Toolbar. Cut Cells The selection will be copied to the clipboard buffer. The selection will be moved to a new location as soon as you choose Paste command as described below. Please note that the selection will remain in its current location until you paste it. If you want to delete a selection without pasting it to a new location, use Delete command instead. Copy Cells The selection will be copied to the clipboard buffer. The selection will be copied to a new location as soon as you choose Paste command as described below. You can copy all the cells of a column or a row selecting the rows or columns and using the COPY CELLS button. To avoid copying the variable names, when you select some columns and click the COPY CELLS button, the name cells of the column is deselected. Paste Cells The contents of the clipboard will be pasted in the spreadsheet so that the selected cell becomes the top left corner of the selection. When you paste cells, the blank cells that you cut/copied will not modify any existing value and will not add any new value to the column. The missing values that you copied will add new missing values or will replace the existing ones with missing values. If you copy a discontinuous group of cells, by selecting cells or block of cells while holding down the Ctrl key, ISSTATS will paste it as a continuous selection where non selected cells are converted in empty strings. Insert Cells You can insert copied cells above of the active cell on your worksheet, shifting other cells in the same column down. After you insert cells, the blank cells that you previously copied will add missing values. Inserting cells below the last value of the column will do as pasting cells: blank cells will not add data. Clear Cells Erases the contents of the selected cells, without moving other cells. This menu command is available when at least one cell is selected in the Data window. To delete selected cells, see Delete Cells. In a numeric column, ISSTATS inserts system missing values (*) in a cleared cell. Delete Cells
11. 11. APPLICATION MANUAL V0.4 InnerSoft STATS 11 If you are deleting cells, other cells automatically shift up. If you click a cell and then press DELETE, Excel deletes the cell contents. Select All Cells Click this button to select all cells on the worksheet. Copy Columns When you copy various columns, alternated selected columns will be joined together in a continuous selection once you paste or insert them (as Excel does). Paste Columns On pasting, empty selected columns will not modify any existing variables. Not empty selected columns will replace any existing variables. Insert Columns You can insert columns that previously copied in the worksheet. To insert a single column, select the column in the column immediately to the right of where you want to insert the new column. To insert multiple columns, select the columns immediately to the right of where you want to insert columns. If you insert columns, other columns automatically shift to the right. On inserting, empty columns (no variable defined) will add new empty columns (no variable defined). Clear Columns Content Erases the contents of the selected column, without moving other columns. This menu command is available when at least one column is selected in the Data window. To delete selected columns, see Delete Columns. The cleared cells remain as blank cells on the worksheet. Delete Columns If you are deleting columns, other columns automatically shift to the left. Select All Columns The Select All Columns command will create a new selection that includes every column with data. As rows have not any structural meaning in the table, there is no copy/cut/paste/insert options for rows. If you want to copy all cells of a row just select the rows and press COPY CELLS button. 4.3. – Options Menu
12. 12. APPLICATION MANUAL V0.4 InnerSoft STATS 12 Table Decimal Places For numbers that are already entered on a worksheet, you can increase or decrease the number of places that are displayed after the decimal point by using the Table Decimal Places buttons. By default, ISSTATS displays 3 decimal places when you apply a built-in number format to the cells. Table Columns You can increase or decrease the number of columns that are available in the worksheet by using the Table Columns buttons. By default, ISSTATS displays 10 columns. To avoid memory problems, limit the number of table columns when operating with huge data. If you want to handle variables with millions of values, set the number of table columns to minimum: if you are using only 4 variables, set only 4 columns. Table Rows You can increase or decrease the number of rows that are available in the worksheet by using the Table Rows buttons. By default, ISSTATS displays 100 rows.
13. 13. APPLICATION MANUAL V0.4 InnerSoft STATS 13 5. – Working in the Output window New Document Creating a new output document closes the current document. If your current document has been changed since it was last saved, ISSTATS will prompt you to save the document before closing it. Open Document Opening a new document closes the current document. If your current document has been changed since it was last saved, ISSTATS will prompt you to save the document before closing it. The file format of the opened document must be ISSTAT XAML package (a XAML package file produced by ISSTAT). Save Document Saves the current document in a ISSTAT XAML package file (a XAML package file produced by ISSTAT). If the document has been saved before, Save Document saves the file with the current file name without displaying a dialog box. If the document has not been saved before, Save Document opens the Save Document As dialog box. Use Windows Clipboard to copy and paste the document content in other applications, such as MS Word, MS Excel, Notepad… Save Document As Saves the document in a file. Use this command if you want to rename your document or save it to a new location. Print Document Prints the document. Cut, Copy & Paste The way to copy or move a selection is to use cut, copy, and paste operations. Use this commands to copy a selection from Output window to other applications, such as MS Word, MS Excel, Notepad… Undo & Redo To undo an action, click Undo Button on the Toolbar. To redo an action that you undid, click Redo Button on the Toolbar. Bold, Italic, Underline, Grow Font, Shrink Font, Bullets, Numbering, Align Left, Align Center, Align Right, Align Justify, Increase Indent & Decrease Indent
14. 14. APPLICATION MANUAL V0.4 InnerSoft STATS 14 These are the formatting buttons, familiar to anyone who has used a word processor. These formats are applied by selecting some text in the window and clicking the button in the editor toolbar. Select All The Select All command will create a new selection that includes everything on the document.
15. 15. APPLICATION MANUAL V0.4 InnerSoft STATS 15 6. – Descriptive Statistics Overview Produces descriptive statistics for each column. The data columns must be numeric and contain at least one numeric value. The dialog box allows to choose the statistics that you wish to display. Available Variables list shows numeric data columns containing at least one no-missing value. Dialog box items Mean: Choose to display the arithmetic mean. Sample Variance: Choose to display the unbiased variance of the data. Estimates population variance based on a sample. If your data represents the entire population, then compute the variance by using Total Variance. Sample Standard Deviation: Choose to display the standard deviation of the data. Estimates population standard deviation based on a sample. If your data represents the entire population, then compute the Std. Deviation by using Total Std. Deviation. Sample Coefficient of variation: Choose to display the coefficient of variation. Sample Skewness: Choose to display the skewness value. Estimates population skewness based on a sample. If your data represents the entire population, then compute the skewness by using Total Skewness. Sample Kurtosis: Choose to display the kurtosis value. Estimates population kurtosis based on a sample. If your data represents the entire population, then compute the kurtosis by using Total Kurtosis.
16. 16. APPLICATION MANUAL V0.4 InnerSoft STATS 16 Total Variance: Choose to display the variance of the data. Calculates variance based on the entire population. Total Standard deviation: Choose to display the standard deviation of the data. Calculates Std. Deviation based on the entire population. Total Coefficient of variation: Choose to display the coefficient of variation. Calculates coefficient of variation based on the entire population. Total Skewness: Choose to display the skewness value. Calculates skewness based on the entire population. Total Kurtosis: Choose to display the kurtosis value. Calculates kurtosis based on the entire population. SEM: Choose to display the standard error of the mean. Sum: Choose to display the data sum. Minimum: Choose to display the data minimum. Maximum: Choose to display the data maximum. Range: Choose to display the data range. Data Range is the difference between the maximum and minimum. Quartiles: Choose to display the first quartile, the median and the third quartile. Interquartile range: Choose to display the difference between the first and third quartiles. Deciles: Choose to display the nine values that divide the sorted data into ten equal parts. Percentiles: To request a percentile:  Enter the desired value in the box placed in Percentiles group box. For example, if you wanted the 7th percentile, you would enter a 7 in the box.  Click the add button to add the percentile to the list of requested percentiles.  Repeat Step 1 and 2 to add additional percentiles as desired.  If you need to delete a percentile, select it in the list and click the remove button. Mode: Choose to display the mode and the number of times it occurs. If multiple modes exist, Minitab displays the smallest modes, up to a total of four, along with their frequency. Sum of squares: Choose to display the sum of the squared data values. This is the uncorrected sums of squares, without first subtracting the mean. MSSD: Choose to display half the Mean of Successive Squared Differences. N nonmissing: Choose to display the number of nonmissing column entries. N missing: Choose to display the number of missing column entries. N total: Choose to display the total (nonmissing and missing) number of column entries.
17. 17. APPLICATION MANUAL V0.4 InnerSoft STATS 17 Cut–Points: Divide the data into a number of equal groups. For example, to create deciles, you would enter 10 in the box .Enter 3 to divide the data into tertiles. Check statistics Check None: Choose to clear all check boxes and then individually check the statistics to display. Check All: Choose to check all boxes. You can uncheck statistics as needed.
18. 18. APPLICATION MANUAL V0.4 InnerSoft STATS 18 7. – One Sample Test Overview In the box Test Type, choose the test to perform:  Z Test  T Test  Variance Test Hypothesis Test To perform a hypothesis test, check the box Perform Hypothesis Test and choose the alternative hypothesis of the test:  Less than: Perform a level  test of H0:  ≥ 0 against the one-sided alternative H1:  < 0  Not Equal: Perform a level  test of H0:  = 0 against the two-sided alternative H1:  ≠ 0  Greater than: Perform a level  test of H0:  ≤ 0 against the one-sided alternative H1:  > 0 Significance level of the test () derive from the 1- value set in the Confidence Level text box. If you choose a lower-tailed hypothesis test, an upper confidence bound will be constructed. If you choose an upper-tailed hypothesis test, a lower confidence bound will be constructed. Available Variables list shows numeric data columns containing at least one no-missing value. 7.1. – Z Test Use 1-Sample Z to compute a confidence interval or perform a hypothesis test of the mean when the standard deviation of the population σ is known. The samples should come from a normal population if n is low; if however n>30 the distribution of the population does not have to be normal.
19. 19. APPLICATION MANUAL V0.4 InnerSoft STATS 19 Dialog box items Data: Choose the origin of the data.  Samples in columns: Choose if you have entered raw data in columns. Enter the columns containing the sample data in the list Computing Variables. Move these variables from Available Variables list to Computing Variables list using Add and Remove buttons. Enter the value for the population standard deviation in the text box. Entering multiple columns, ISSTATS performs separate one-sample analyses on each column.  Summarized data: Choose if you have summary values for the sample size and mean. o Sample size: Enter the value for the sample size. o Mean: Enter the value for the sample mean. o Population Standard deviation: Enter the value for the population standard deviation. Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9 will result in a 90% confidence interval. The default is 0,95 = 95%. Perform hypothesis test: Check to perform the hypothesis test.  Hypothesized mean: Enter the test mean 0.  Alternative hypothesis: Choose the alternative hypothesis of the test. 7.2. – T Test Performs a one-sample t-test or t-confidence interval for the mean. Use T Test for one sample to compute a confidence interval and perform a hypothesis test of the mean when the population standard deviation, σ, is unknown. Use this test when samples come from a normal population or n > 30. Dialog box items Data: Choose the origin of the data.  Samples in columns: Choose if you have entered raw data in columns. Enter the columns containing the sample data in the list Computing Variables. Move these variables from Available Variables list to Computing Variables list using Add and Remove buttons. Entering multiple columns, ISSTATS performs separate one-sample analyses on each column  Summarized data: Choose if you have summary values for the sample size and mean. o Sample size: Enter the value for the sample size. o Mean: Enter the value for the sample mean. o Sample Standard deviation: Enter the value for the sample standard deviation. Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9 will result in a 90% confidence interval. The default is 0,95 = 95%.
20. 20. APPLICATION MANUAL V0.4 InnerSoft STATS 20 Perform hypothesis test: Check to perform the hypothesis test.  Hypothesized mean: Enter the test mean 0.  Alternative hypothesis: Choose the alternative hypothesis of the test. 7.3. – Variance Test This procedure calculates confidence intervals for the variance of a population, and performs a hypothesis test to determine whether the population variance equals a specified value. Use this test when samples come from a normal population. Dialog box items Data: Choose the origin of the data.  Samples in columns: Choose if you have entered raw data in columns. Enter the columns containing the sample data in the list Computing Variables. Move these variables from Available Variables list to Computing Variables list using Add and Remove buttons. Entering multiple columns, ISSTATS performs separate one-sample analyses on each column.  Summarized data: Choose if you have summary values for the sample size and variance. o Sample size: Enter the value for the sample size. o Sample Variance: Enter the value for the sample variance. Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9 will result in a 90% confidence interval. The default is 0,95 = 95%. Perform hypothesis test: Check to perform the hypothesis test.  Hypothesized variance: Enter the test variance σ2 0.  Alternative hypothesis: Choose the alternative hypothesis of the test.
21. 21. APPLICATION MANUAL V0.4 InnerSoft STATS 21 8. – Two-Sample Test In the box Test Type, choose the test to perform:  2-Sample t-Test  Paired t-Test  2 Variances F-Test Hypothesis Test To perform a hypothesis test, check the box Perform Hypothesis Test and choose the Alternative Hypothesis of the test:  Less than: Perform a level  test of H0:  ≥ d against the one-sided alternative H1:  < d  Not Equal: Perform a level  test of H0:  = d against the two-sided alternative H1:  ≠ d  Greater than: Perform a level  test of H0:  ≤ d against the one-sided alternative H1:  > d Significance level of the test () derive from the 1- value set in the Confidence Level text box. If you choose a lower-tailed hypothesis test, an upper confidence bound will be constructed. If you choose an upper-tailed hypothesis test, a lower confidence bound will be constructed. Available Variables list shows numerical and text columns containing at least one no-missing value. Text columns may be used as Subscripts. 8.1. – 2 Sample t-Test Performs an independent 2-sample t-test and generates a confidence interval.
22. 22. APPLICATION MANUAL V0.4 InnerSoft STATS 22 When you have dependent samples, use 8.2. – Paired t-Test. Use 2-Sample t-Test to perform a hypothesis test and compute a confidence interval of the difference between two population means when the population standard deviations, σ's, are unknown. Dialog box items Data: Choose the origin of the data.  Samples in one column: Choose if the sample data are in a single column, differentiated by subscript values (group codes) in a second column. o Computing Variables: Enter the columns containing the sample data in the list Computing Variables. Move these variables from Available Variables list to Computing Variables list using Add / Remove buttons. o Grouping Variable: Enter the column containing the sample subscripts to define the groups. It may be a numerical or text column. o Group 1/Group 2: Enter the subscripts that define both groups. o Automatic Grouping: Check to let the application define automatically both groups. It will identify the first two different subscripts in the grouping variable.  Samples in different columns: Choose if the data of the two samples are in separate columns. o First Sample: Enter the column containing one sample from Available Variables list. o Second Sample: Enter the column containing the other sample from Available Variables list.  Summarized data: Choose if you have summary values for the sample size, mean, and variance for each sample. o First Sample  Sample size 1: Enter the sample size for the first sample.  Mean 1: Enter the value for the mean of the first sample.  Sample Variance 1: Enter the value for the variance of the first sample. o Second Sample  Sample size 2: Enter the sample size for the second sample.  Mean 2: Enter the value for the mean of the second sample.  Sample Variance 2: Enter the value for the variance of the second sample. Assume Population equal variances: Check to assume that the populations have equal variances. Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9 will result in a 90% confidence interval. The default is 0,95 = 95%. Perform hypothesis test: Check to perform the hypothesis test.  Hypothesized mean: Enter the hypothesized difference between the two population means 1-2.  Alternative hypothesis: Choose the alternative hypothesis of the test. A difference of 0 suggests in the null hypothesis the equality between mean populations; H0: 1-2 = 0 against an alternative H1: 1-2 ≠ 0
23. 23. APPLICATION MANUAL V0.4 InnerSoft STATS 23 A difference equal or less than 0 suggest in the null hypothesis that the first mean is equal or less than the second; H0: 1-2 ≤ 0 against an alternative H1: 1-2 > 0 A difference equal or greater than 0 suggest in the null hypothesis that the first mean is equal or greater than the second; H0: 1-2 ≥ 0 against an alternative H1: 1-2 < 0 Optionally, test ratios other than 0 (equality) can be specified. A difference of 2 suggests in the null hypothesis the first mean is the second mean plus 2. 8.2. – Paired t-Test Performs a paired t-test. This is appropriate for testing the mean difference between paired observations when the paired differences follow a normal distribution. Use the Paired t command to compute a confidence interval and perform a hypothesis test of the mean difference between paired observations in the population. A paired t-test matches responses that are dependent or related in a pairwise manner. A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication. Paired samples t-tests are often referred to as "dependent samples t-tests". When the samples are drawn independently from two populations, use 8.1. – 2 Sample t-Test. Dialog box items Data: Choose the origin of the data.  Sample in columns: Choose if you have entered raw data in two columns. o First sample: Enter the column containing the first sample from Available Variables list. o Second sample: Enter the column containing the second sample from Available Variables list. Pairs must have two numerical values. Pairs that have a missing data in any of the members are ignored.  Summarized data: Choose if you have summary values for the sample size, mean, and variance of the difference. o Sample size: Enter the value for the sample size. o Mean of Differences: Enter the value for the mean of differences ̅. o Variance of Differences: Enter the value for the variance of differences s2 d. Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9 will result in a 90% confidence interval. The default is 0,95 = 95%. Perform hypothesis test: Check to perform the hypothesis test.  Hypothesized mean: Enter the hypothesized population mean of the paired differences d.  Alternative hypothesis: Choose the alternative hypothesis of the test. 8.3. – 2 Variances F-Test
24. 24. APPLICATION MANUAL V0.4 InnerSoft STATS 24 The 2 Variances F-Test procedure performs hypothesis tests and computes confidence intervals for the ratios between two populations' variances. Use this test to determine if one treatment condition has more variability than the other. Each population must follow the normal distribution. Dialog box items Data: Choose the origin of the data.  Samples in one column: Choose if you have entered data into a single column with a second column of subscripts that identify the samples. o Computing Variables: Enter the columns containing the sample data in the list Computing Variables. Move these variables from Available Variables list to Computing Variables list using Add and Remove buttons. Entering multiple columns, ISSTATS performs separate analyses on each column. o Grouping Variable: Enter the column containing the sample subscripts to define the groups. It may be a numerical or text column. o Group 1/Group 2: Enter the subscripts that define both groups. o Automatic Grouping: Check to let the application define automatically both groups. It will identify the first two different subscripts in the grouping variable.  Samples in different columns: Choose if you have entered the data for the two samples into separate columns. o First Sample: Enter the column that contains the data for the first sample from Available Variables list. o Second Sample: Enter the column that contains the data for the second sample from Available Variables list.  Summarized data: Choose if you have summary values for the sample sizes and variances. o First Sample  Sample size 1: Enter the sample size for the first sample.  Sample Variance 1: Enter the variance for the first sample. o Second Sample  Sample size 2: Enter the sample size for the second sample.  Sample Variance 2: Enter the variance for the second sample. Confidence level: Enter the level of confidence desired. Enter any number between 0 and 1. Entering 0,9 will result in a 90% confidence interval. The default is 0,95 = 95%. Perform hypothesis test: Check to perform the hypothesis test.  Hypothesized ratio: Enter the hypothesized ratio between two population’s variances σ2 1/σ2 2.  Alternative hypothesis: Choose the alternative hypothesis of the test. A ratio of 1 suggests in the null hypothesis the equality between variance populations; H0: σ2 1/σ2 2 = 1 against an alternative H1: σ2 1/σ2 2 ≠ 1. A ratio equal or less than 1 suggest in the null hypothesis that the first variance is equal or less than the second; H0: σ2 1/σ2 2 ≤ 1 against an alternative H1: σ2 1/σ2 2 > 1.
25. 25. APPLICATION MANUAL V0.4 InnerSoft STATS 25 A ratio equal or greater than 1 suggest in the null hypothesis that the first variance is equal or greater than the second; H0: σ2 1/σ2 2 ≥ 1 against an alternative H1: σ2 1/σ2 2 < 1. Optionally, test ratios other than 1 (equality) can be specified. A ratio of 2 suggests the first variance is double the second variance.
26. 26. APPLICATION MANUAL V0.4 InnerSoft STATS 26 9. – One-Way ANOVA Performs a one-way analysis of variance. You can also perform multiple comparisons. The response variable must be numeric. The factor level column (grouping variable) may be numeric or text. Dialog box items Data: Choose the origin of the data.  Groups in different Columns o Computing Variables: Enter the columns containing the response. Each column must contain the data for one of the groups.  Groups in 1 Column o Computing Variables: Enter the column or columns containing the response. ISSTATS performs a ANOVA test for each of these columns. o Grouping Variable: Enter the column containing the factor levels. Confidence level: Enter the confidence level. For further details, see ANOVA Test. Multiple Comparisons: Use to generate grouping information tables and confidence intervals for the differences between means, by different methods. Check to obtain confidence intervals for all pairwise differences between level means using any of these methods.  Scheffe  Tukey HSD: Tukey's Honestly Significant Difference test.  Sidak
27. 27. APPLICATION MANUAL V0.4 InnerSoft STATS 27  Fisher LSD: Fisher's Least Significant Difference test.  Bonferroni For further details, see ANOVA Multiple Comparisons. Include Brown-Forsythe Test for equality of Means Use to test the equality of means when distribution violates the assumption of equal variances. The Brown-Forsythe test cannot be computed if all groups have zero variance. To take part into Brown– Forsythe test, a group must have at least two elements. In the situation that some groups have zero standard deviations, the statistic can be computed but the approximation may not work. For further details, see Brown–Forsythe Test for equality of means. Include Welch’s Test for equality of Means Use to test the equality of means when distribution violates the assumption of equal variances. To take part into Welch’s test, a group must have non zero variance. Moreover, sample sizes of a group have to be greater than or equal to 2. For further details, see Welch’s Test for equality of means. Remark: ISSTATS does not cancel a test if any of the groups does not fulfill the conditions. It simply rejects the group. To take part into a ANOVA test, a group must have at least one element. To take part into a Welch’s Test, a group must have at least two elements; moreover, the group must have a non-zero variance. Thus, ISSTATS may use 4 of the groups to perform ANOVA Test but only 2 of the groups to perform a Welch’s Test. It depends on the number of elements of each group. You should read the Descriptive Statistics information to check the Rejected Groups and also should read the Total Groups info of each test to check the number of groups being used in that test. Outputs Outputs include:  Descriptive statistic for every level of the factor’s variable.  Total mean.  Degrees Of Freedom: DFTotal, DFInter and DFIntra  The Sum Of The Squares: SSTotal, SSInter and SSIntra  The Mean Square: MSTotal, MSInter and MSIntra  The F ratio  The p-value  R-squared  R-squared adjusted Inter is also referred as Between Groups or Between Treatments. Intra is also referred as Within Groups or Error Term.
28. 28. APPLICATION MANUAL V0.4 InnerSoft STATS 28 R-squared represents the percentage of variation in a response variable that is explained by its relationship with one predictor variable. R-squared adjusted is a version of r-squared that has been adjusted for the number of predictors in the model. R-squared tends to overestimate the strength of the association, especially when there are more than one independent variables.
29. 29. APPLICATION MANUAL V0.4 InnerSoft STATS 29 10. – Homoscedasticity Tests These tests are used to test if k samples are from populations with equal variances. The response variable has to be numeric. The factor level column (grouping variable) may be numeric or text. Dialog box items In the box Test Type, choose the test to perform:  Levene's Test  Brown–Forsythe Test for equality of variances  Bartlett's Test For further details, see Homoscedasticity Tests. Remark: ISSTATS does not cancel a test if any of the groups does not fulfill the conditions. It simply rejects the group. To take part into a Levene or Brown–Forsythe test, a group must have at least one element. To take part into a Bartlett's Test, a group must have at least two elements; moreover, the group must have a non-zero variance. Thus, ISSTATS may use 4 of the groups to perform Levene's Test but only 2 of the groups to perform a Bartlett's Test. It depends on the number of elements of each group. You should read the Descriptive Statistics information to check the Rejected Groups and also should read the Total Groups info of each test to check the number of groups being used in that test.
30. 30. APPLICATION MANUAL V0.4 InnerSoft STATS 30 11. – Methods and Formulas Help Mean The arithmetic mean is the sum of a collection of numbers divided by the number of numbers in the collection. Sample Variance The estimator of population variance, also called the unbiased sample variance, is: ∑ ( ̅) Source: http://en.wikipedia.org/wiki/Variance Sample Kurtosis The estimators of population kurtosis is: ( ) ( )( )( ) ∑ ( ̅) ( ) ( )( ) The standard error of the sample kurtosis of a sample of size n from the normal distribution is: √ ( ) ( ) ( )( )( )( )( ) Source: http://en.wikipedia.org/wiki/Kurtosis#Estimators_of_population_kurtosis Sample Skewness Skewness of a population sample is estimated by the adjusted Fisher–Pearson standardized moment coefficient: ( )( ) ∑ ( ̅ ) where n is the sample size and s is the sample standard deviation. The standard error of the skewness of a sample of size n from a normal distribution is: √ ( ) ( )( )( ) Source: https://en.wikipedia.org/wiki/Skewness#Sample_skewness Total Variance
31. 31. APPLICATION MANUAL V0.4 InnerSoft STATS 31 Variance of the entire population is: ∑ ( ̅) Source: http://en.wikipedia.org/wiki/Variance Total Kurtosis Kurtosis of the entire population is: ∑ ( ̅) where n is the sample size and σ is the total standard deviation. Source: http://en.wikipedia.org/wiki/Kurtosis Total Skewness Skewness of the entire population is: ∑ ( ̅) where n is the sample size and σ is the total standard deviation. Source: https://en.wikipedia.org/wiki/Skewness Quantiles of a population ISSTATS uses the same method as R–7, Excel CUARTIL.INC function, SciPy–(1,1), SPSS and Minitab. Qp, the estimate for the kth q–quantile, where p = k / q and h = (N–1)*p + 1, is computing by Qp = Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1]. When p = 1, use xN. Source: http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population MSSD (Mean of the squared successive differences) It is calculated by taking the sum of the differences between consecutive observations squared, then taking the mean of that sum and dividing by two. ∑ ( ) ( ) The MSSD has the desirable property that one half the MSSD is an unbiased estimator of true variance.
32. 32. APPLICATION MANUAL V0.4 InnerSoft STATS 32 Confidence Bounds and One-Sided Tests Suppose you are testing the null hypothesis H0:  ≥ 0 against the one-sided alternative H1:  < 0. Rather than give a two-sided confidence interval for , the more appropriate procedure is to give an upper confidence bound in this setting. This upper confidence bound has a direct relationship to the one-sided test, namely: 1. A level  test of H0:  ≥ 0 against the one-sided alternative H1:  < 0 rejects H0 exactly when the value 0 is above the 1–α upper confidence bound. 2. A level  test of H0:  ≤ 0 against the one-sided alternative H1:  > 0 rejects H0 exactly when the value 0 is above the 1–α lower confidence bound. ANOVA Test ∑ ∑( ̅) ∑ (̅ ̅) ∑ ∑( ̅ ) DFTotal = N – 1 DFInter = k – 1 DFIntra = N – k where  F is the result of the test  k is the number of different groups to which the sampled cases belong  ∑ is the total sample size  ni is the number of cases in the i-th group  yij is the value of the measured variable for the j-th case from the i-th group
33. 33. APPLICATION MANUAL V0.4 InnerSoft STATS 33  ̅ is the mean of all yij  ̅ is the mean of the yij for group i. The test statistic has a F-distribution with DFInter and DFIntra degrees of freedom. Thus the null hypothesis is rejected if ( ) ANOVA Multiple Comparisons Difference of Means ̅ ̅ Standard Error of the Difference of Means Estimator √ ( ) Scheffe’s Method Confidence Interval for Difference of Means ( ) ̅ ̅ √ ( ) ( ) Source: http://en.wikipedia.org/wiki/Scheff%C3%A9%27s_method Tukey's range test HSD Confidence Interval for Difference of Means ( ) ̅ ̅ ( ) √ ( ) Where q is the studentized range distribution. Source: https://en.wikipedia.org/wiki/Tukey%27s_range_test Fisher's Method LSD If overall ANOVA test is not significant, you must not consider any results of Fisher test, significant or not. Confidence Interval for Difference of Means ( ) ̅ ̅ ( ⁄ ) √ ( ) Where t is the student distribution.
34. 34. APPLICATION MANUAL V0.4 InnerSoft STATS 34 Bonferroni's Method The family-wise significance level (FWER) is α = 1 - Confidence Level. Thus any comparison flagged by ISSTATS as significant is based on a Bonferroni Correction: ( ) ( ) Where k is the number of groups. Confidence Interval for Difference of Means ( ) ̅ ̅ ( ⁄ ) √ ( ) Where t is the student distribution. Sidak's Method The family-wise significance level (FWER) is α = 1 - Confidence Level. So any comparison flagged by ISSTATS as significant is based on a Sidak Correction: ( ) ( ) ( ) ( ) Where k is the number of groups. Confidence Interval for Difference of Means ( ) ̅ ̅ ( ⁄ ) √ ( ) Where t is the student distribution. Welch’s Test for equality of means The test statistic, F* , is defined as follows: ∑ ( ̅ ̃) ( ) ∑ where
35. 35. APPLICATION MANUAL V0.4 InnerSoft STATS 35  F* is the result of the test  k is the number of different groups to which the sampled cases belong  ni is the number of cases in the i-th group   ∑ ∑  ̃ ∑ ̅  ( ) The test statistic has approximately a F-distribution with k-1 and ∑ degrees of freedom. Thus the null hypothesis is rejected if ( ) Brown–Forsythe Test for equality of means The test statistic, F* , is defined as follows: ∑ ( ̅ ̅ ) ∑ ( ) where  F* is the result of the test  k is the number of different groups to which the sampled cases belong  ni is the number of cases in the i-th group (sample size of group i)  ∑ is the total sample size  ̅ ∑ ̅ is the overall mean. The test statistic has approximately a F-distribution with k-1 and df degrees of freedom. Where df is obtained with the Satterthwaite (1941) approximation as ∑ with ( ) ∑ ( ) Thus the null hypothesis is rejected if ( ) Homoscedasticity Tests Levene's Test The test statistic, F, is defined as follows:
36. 36. APPLICATION MANUAL V0.4 InnerSoft STATS 36 ∑ ( ̅ ̅ ) ∑ ∑ ( ̅ ) where  F is the result of the test  k is the number of different groups to which the sampled cases belong  ∑ is the total sample size  ni is the number of cases in the i-th group  Yij is the value of the measured variable for the j-th case from the i-th group  | ̅ | where ̅ is a mean of i-th group  ̅ is the mean of all Zij  ̅ is the mean of the Zij for group i. The test statistic has a F-distribution with k-1 and N-k degrees of freedom. Thus the null hypothesis is rejected if ( ) Source: http://en.wikipedia.org/wiki/Levene%27s_test Brown–Forsythe Test for equality of variances The test statistic, F, is defined as follows: ∑ ( ̅ ̅ ) ∑ ∑ ( ̅ ) where  F is the result of the test  k is the number of different groups to which the sampled cases belong  ∑ is the total sample size  ni is the number of cases in the i-th group  Yij is the value of the measured variable for the j-th case from the i-th group  | ̃ | where ̃ is a median of i-th group  ̅ is the mean of all Zij  ̅ is the mean of the Zij for group i. The test statistic has a F-distribution with k-1 and N-k degrees of freedom. Thus the null hypothesis is rejected if ( ) Source: http://en.wikipedia.org/wiki/Levene%27s_test Bartlett's Test Bartlett's test is used to test the null hypothesis, H0 that all k population variances are equal against the alternative that at least two are different. If there are k samples with size ni and sample variances S2 i then Bartlett's test statistic is
37. 37. APPLICATION MANUAL V0.4 InnerSoft STATS 37 ( ) ( ) ∑ ( ) ( ) ( ) (∑ ( ) ) where  ∑ is the total sample size  ∑ ( ) is the pooled estimate for the variance The test statistic has approximately a chi-squared distribution with k-1 degrees of freedom. Thus the null hypothesis is rejected if ( ). Source: http://en.wikipedia.org/wiki/Bartlett%27s_test