{"text_format":"1","workbook":"xlsx","n_sheets":9,"sheets":["WELCOME","One sample","Sensory test","Independent samples","Chi2 test","Dependent samples","Thurstone","Help","Calculation"],"not_rendered":{"charts":0,"images":27,"macros":1,"external_links":0,"defined_names":0,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"WELCOME","index":0,"n_rows":25,"n_cols":26,"rows_kept":25,"cut":false,"hidden":false,"merged":[[0,0,0,14],[16,0,16,14],[18,0,18,14]]}
["Welcome to the V-Power program\nA power tool for sensory analysis"]
[]
[null,null,null,null,"1) Press the START button and select the appropriate test"]
[]
[null,null,null,null,"2) Everytime you want to change the type of test, come back to this page and select another test"]
[]
[]
[]
[]
[]
[]
[null,null,null,null,"3) Help with the procedure?                Click here-->"]
[]
[null,null,null,null,"4) Calculation details?                Click here-->"]
[]
[]
["V-power has been written by Virginie Jesionka and is based on the the Discrimination Test Planning and Analysis Tools developed by Tom Carr (Carr Consulting, 1215 Washington Ave. Suite 203 Wilmette, IL 60091 USA)"]
["Contact : Virginie Jesionka  +33 (0) 6 70 23 06 05  virginiejesionka@gmail.com"]
["Disclaimer\nThe authors have carefully developed this software tool in accordance with their current knowledge. Use of this software is at the user's own risk. Damage and warranty claims arising from missing or incorrect data are excluded. The authors bear no responsibility or liability for damage of any kind, also for indirect or consequential damages resulting use of this software."]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Triangle"]
[]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"th"]

{"sheet":"One sample","index":1,"n_rows":26,"n_cols":22,"rows_kept":26,"cut":false,"hidden":true,"merged":[[24,10,24,15],[22,10,22,15],[3,17,3,23],[6,17,6,23],[5,17,5,23],[4,9,4,15],[17,2,17,7],[22,2,22,7],[9,17,9,23],[10,17,10,23],[19,10,19,15],[10,1,10,7],[21,10,21,15],[17,10,17,15],[0,0,0,16],[1,0,1,16],[3,9,3,15],[3,1,3,7],[4,1,4,7]]}
["Comparison of means _ One sample"]
[]
[]
[null,"BEFORE the test",null,null,null,null,null,null,null,"AFTER the test",null,null,null,null,null,null,null,"Interpretation"]
[null,"You are looking for the sample size",null,null,null,null,null,null,null,"You are looking for the parameters of the test"]
[null,null,"standard deviation (σ)",null,null,null,null,null,null,null,"N"]
[]
[]
[null,null,"Power",null,null,null,null,null,null,null,"Delta for power (D)"]
[null,null,null,null,null,null,null,null,null,null,"α (alpha)"]
[null,"You are looking for the power"]
[null,null,"N"]
[null,null,"standard deviation (σ)",null,null,"Power"]
[null,null,"delta (D)"]
[null,null,"α (alpha)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"               For further information:"]
[]
[null,null,"α (alpha) : type I error, risk of being wrong when saying the mean is different",null,null,null,null,null,null,null,"α (alpha) : type I error, risk of being wrong when saying the mean is different"]
[null,null,"Delta (D) : Minimum difference that is wished to be able to notice",null,null,null,null,null,null,null,"Delta mesured (Dm) : Difference between the sample mean and the theorical value"]
[null,null,"N : Sample size",null,null,null,null,null,null,null,"Delta for power (D) : Minimum difference that is wished to be able to notice"]
[null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products",null,null,null,null,null,null,null,"N : Sample size"]
[null,null,"are similar",null,null,null,null,null,null,null,"p-value : Equivalent to alpha (type I error), once the test has been run"]
[null,null,"Standard deviation (σ) : Can be either the theorical standard deviation or the calculated ",null,null,null,null,null,null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products"]
[null,null,"standard deviation of the sample",null,null,null,null,null,null,null,"are similar"]
[null,null,null,null,null,null,null,null,null,null,"Standard deviation (σ) : Can be either the theorical standard deviation or the calculated "]
[null,null,null,null,null,null,null,null,null,null,"standard deviation of the sample"]

{"sheet":"Sensory test","index":2,"n_rows":24,"n_cols":25,"rows_kept":24,"cut":false,"hidden":true,"merged":[[0,0,0,18],[1,0,1,18],[6,14,7,14],[3,1,3,7],[3,9,3,17],[4,1,4,7],[4,9,4,17],[5,14,5,15],[20,10,21,17],[17,2,18,7],[4,19,4,26],[16,10,17,17],[9,9,9,17],[9,1,9,7]]}
["Duo-Trio test calculator"]
["Those results are effective in the case of N independent duo-trio tests. If the tests include repetitions, check this paragraph :"]
[]
[null,"BEFORE the test",null,null,null,null,null,null,null,"AFTER the test",null,null,null,null,null,null,null,null,null,null,null,"  INTERPRETATION"]
[null,"You are looking for the sample size",null,null,null,null,null,null,null,"You are looking for the proportion of distinguishers",null,null,null,null,null,null,null,null," "]
[null,null,"pD",null,null,null,null,null,null,null,"N",null,null,null,"pD (ND)"]
[null,null,"α (alpha)",null,null,"N",null,null,null,null,"NC",null,null,null,"Confidence interval of pD","pDmin"]
[null,null,"Power",null,null,null,null,null,null,null,"Confidence level",null,null,null,null,"pDmax"]
[null,null,null,null,null,null,0.8]
[null,"You are looking for the power",null,null,null,null,null,null,null,"You are looking for the parameters of the test"]
[null,null,"pD",null,null,null,null,null,null,null,"N"]
[null,null,"α (alpha)",null,null,"Power",null,null,null,null,"NC",null,0.33333333333333331,null,"p-value"]
[null,null,"N",null,null,null,null,null,null,null,"pD",null,null,null,"Power"]
[null,null,null,null,null,null,null,null,null,null,"α (alpha)"]
[]
[]
[null,null,"α (alpha) : type I error, risk of being wrong when saying the products are different",null,null,null,null,null,null,null,"Confidence interval of pD : Min and Max values of the confidence interval. The percentage of confidence is given by the confidence level"]
[null,null,"pD : Proportion of distinguishers the user is willing to accept (above which products are considered different)"]
[null,null,null,null,null,null,null,null,null,null,"N : Sample size"]
[null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the",null,null,null,null,null,null,null,"NC : Number of people who had a correct answer (can be correct out of luck)"]
[null,null,"products are similar",null,null,null,null,null,null,null,"pD (ND) : Proportion of people who distinguished the products (Number of people who distinguished the products)"]
[null,null,"N : Sample size"]
[null,null,null,null,null,null,null,null,null,null,"p-value : Equivalent to alpha (type I error), once the test has been run",null,null,null,null,null,null,null,null,null,null,null,null,null,"For further information:"]
[null,null,null,null,null,null,null,null,null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products are similar"]

{"sheet":"Independent samples","index":3,"n_rows":29,"n_cols":30,"rows_kept":29,"cut":false,"hidden":false,"merged":[[10,24,10,32],[3,24,3,32],[5,24,5,32],[6,24,6,32],[7,24,7,32],[9,24,9,32],[0,0,0,20],[11,1,11,10],[4,1,4,10],[3,12,3,22],[4,12,4,22],[1,0,1,20],[3,1,3,10]]}
["Comparison of means for independent samples"]
["Those results are effective in the case of independent answers. If the test includes repetitions, the results must be considered as approximations."]
[]
[null,"BEFORE the test",null,null,null,null,null,null,null,null,null,null,"AFTER the test",null,null,null,null,null,null,null,null,null,null,null,"INTERPRETATION"]
[null,"You are looking for the sample size",null,null,null,null,null,null,null,null,null,null,"You are looking for the parameters of the test"]
[null,null,"Standard deviation (σ)",null,"or","σ1",null,null,null,null,null,null,null,"N",null," or","n1"]
[null,null,null,null,null,"σ2",null,null,null,null,null,null,null,null,null,null,"n2"]
[null,null,"Delta (D)",null,null,null,null,null,null,null,null,null,null,"Standard deviation (σ)",null," or","σ1"]
[null,null,"α (alpha)",null,null,null,null,null,"N",null,null,null,null,null,null,null,"σ2"]
[null,null,"Power",null,null,null,null,null,null,null,null,null,null,"Delta mesured (Dm)",null,null,null,null,null,null,"p-value"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Delta for power (D)",null,null,null,null,null,null,"Power"]
[null,"You are looking for the power",null,null,null,null,null,null,null,null,null,null,null,"α (alpha)"]
[null,null,"N",null,"or","n1"]
[null,null,null,null,null,"n2"]
[null,null,"Standard deviation (σ)",null,"or","σ1"]
[null,null,null,null,null,"σ2",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"            For further information :"]
[null,null,"Delta (D)",null,null,null,null,null,"Power"]
[null,null,"α (alpha)"]
[]
[]
[null,"α (alpha) : Type I error, risk of being wrong when saying the means are different",null,null,null,null,null,null,null,null,null,null,null,"α (alpha) : Type I error, risk of being wrong when saying the means are different"]
[null,"Delta (D) : Difference between the means of the two samples above which the products are considered different",null,null,null,null,null,null,null,null,null,null,null,"Delta mesured (Dm) : Actual difference between the means of the two samples"]
[null,"N : Total samples size (sum of two samples) N = n1 + n2",null,null,null,null,null,null,null,null,null,null,null,"Delta for power (D) : Difference between the groups' means above which the products are considered different"]
[null,"     n1 and n2 : Size of each sample (necessary if the size is different). N = n1 + n2",null,null,null,null,null,null,null,null,null,null,null,"N : Total samples size (sum of two samples) N = n1 + n2"]
[null,"Power : Power of the test, 1- type II error, chance of being right when saying the products are similar",null,null,null,null,null,null,null,null,null,null,null,"     n1 and n2 : Size of each sample (necessary if the size is different). N = n1 + n2"]
[null,"σ : Standard deviation of samples or an estimation of it",null,null,null,null,null,null,null,null,null,null,null,"p-value : Equivalent to alpha (type I error), once the test has been run"]
[null,"     σ1 and σ2 : Standard deviation of each sample (necessary if they are different*).  See * in the \"Help\" worksheet )",null,null,null,null,null,null,null,null,null,null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products are similar"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"σ : Standard deviation of samples or an estimation of it"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"     σ1 and σ2 : Standard deviation of each sample (necessary if they are different*).  See * in the \"Help\" worksheet )"]

{"sheet":"Chi2 test","index":4,"n_rows":21,"n_cols":24,"rows_kept":21,"cut":false,"hidden":true,"merged":[[0,0,0,15],[11,11,11,12],[5,11,5,12],[3,1,3,7],[4,1,4,7],[9,1,9,7],[3,9,3,17],[4,9,4,17],[7,9,8,9],[1,0,1,15],[18,2,18,7],[17,9,17,17],[16,2,16,7],[15,2,15,7],[19,9,19,12]]}
["Same/Different test calculator"]
["Those results are effective in the case of N independent answers. If the test includes repetitions, check this paragraph :"]
[]
[null,"BEFORE the test",null,null,null,null,null,null,null,"AFTER the test",null,null,null,null,null,null,null,null,null,null,null," INTERPRETATION"]
[null,"You are looking for the sample size",null,null,null,null,null,null,null,"You are looking for the parameters of the test"]
[null,null,"effet size φ",null,null,null,null,null,null,null,null,"Subjects RECEIVED"]
[null,null,"α (alpha)",null,null,"N",null,null,null,null,null,"Matched pairs","Unmatched pairs","Total",null,"effect size φ"]
[null,null,"Power",null,null,null,null,null,null,"Subjects SAID","Same",null,null,48,null,"χ2 value"]
[null,null,null,null,null,null,0.8,null,null,null,"Different",null,null,61,null,"p-value"]
[null,"You are looking for the power",null,null,null,null,null,null,null,null,"Total",39,70,109,null,"λ"]
[null,null,"effect size φ",null,null,null,null,null,null,null,null,null,null,null,null,"Power"]
[null,null,"α (alpha)",null,null,"Power",null,null,null,null,null," Level of SIGNIFICANCE for the p-value"]
[null,null,"N",null,null,null,null,null,null,null,null,"α",0.2]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"For furter information:"]
[null,null,"N : Sample size",null,null,null,null,null,null,"  α : Type I error, significance level for the p-value"]
[null,null,"Effect size phi (φ) : φ=0.1: small, φ=0.3: medium and φ=0.5: large effect, by",null,null,null,null,null,null,"  χ2 : Chi Square parameter = φ2*N"]
[null,null,"convention",null,null,null,null,null,null,"  Effect size phi (φ) : Actual mesure based on the data"]
[null,null,"N : Sample size",null,null,null,null,null,null,"  λ : Non-centrality parameter"]
[null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products",null,null,null,null,null,null,"  p-value : Equivalent to alpha (type I error), once the test has been run"]
[null,null,"are similar",null,null,null,null,null,null,"  Power : Power of the test, 1- type II error, chance of being right when saying the products are similar"]

{"sheet":"Dependent samples","index":5,"n_rows":30,"n_cols":30,"rows_kept":30,"cut":false,"hidden":true,"merged":[[0,0,0,20],[3,24,3,32],[4,1,4,10],[4,12,4,22],[12,1,12,10],[1,0,1,20],[3,1,3,10],[3,12,3,22],[5,24,5,32],[6,24,6,32],[7,24,7,32],[9,24,9,32],[10,24,10,32]]}
["Comparison of means for dependent samples "]
["Those results are effective in the case of independent paired answers. If the test includes repetitions, the results must be considered as approximations."]
[]
[null,"BEFORE the test",null,null,null,null,null,null,null,null,null,null,"AFTER the test",null,null,null,null,null,null,null,null,null,null,null,"INTERPRETATION"]
[null,"You are looking for the sample size",null,null,null,null,null,null,null,null,null,null,"You are looking for the parameters of the test"]
[null,null,"standard deviation (σ)",null,"or","σ1",null,null,null,null,null,null,null,"N"]
[null,null,null,null,null,"σ2",null,null,null,null,null,null,null,"standard deviation (σ)",null," or","σ1"]
[null,null,"Delta (D)",null,null,null,null,null,null,null,null,null,null,null,null,null,"σ2"]
[null,null,"α (alpha)",null,null,null,null,null,null,null,null,null,null,"Delta mesured (Dm)"]
[null,null,"Power",null,null,null,null,null,"N",null,null,null,null,"Delta for power (D)",null,null,null,null,null,null,"p-value"]
[null,null,"correlation coefficient r(1,2)",null,null,null,null,null,null,null,null,null,null,"correlation coefficient r(1,2)",null,null,null,null,null,null,"Power"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"α (alpha)"]
[null,"You are looking for the power"]
[null,null,"N"]
[null,null,"standard deviation (σ)",null,"or","σ1"]
[null,null,null,null,null,"σ2",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"             For further information :"]
[null,null,"Delta (D)"]
[null,null,"α (alpha)",null,null,null,null,null,"Power"]
[null,null,"correlation coefficient r1,2"]
[]
[]
[null,"α (alpha) : Type I error, risk of being wrong when saying the means are different",null,null,null,null,null,null,null,null,null,null,null,"α (alpha) : Type I error, risk of being wrong when saying the means are different"]
[null,"correlation coefficient r1,2 : between the two samples (0 independent, 1 perfectly correlated)",null,null,null,null,null,null,null,null,null,null,null,"correlation coefficient r1,2 : between the two samples (0 independent, 1 perfectly correlated)"]
[null,"Delta (D) : Difference between the groups' means or Mean of the differences between each sample pair",null,null,null,null,null,null,null,null,null,null,null,"Delta mesured (Dm) : Actual difference between the groups' means or Mean of the differences between each sample pair"]
[null,"N : Sample size, number of pairs of observations",null,null,null,null,null,null,null,null,null,null,null,"Delta (D) : Difference between the groups' means above which products are considered differents"]
[null,"Power : Power of the test, 1- type II error, chance of being right when saying the products are similar",null,null,null,null,null,null,null,null,null,null,null,"N : sample size, number of pairs of observations"]
[null,"σ : Standard deviation of samples or an estimation of it",null,null,null,null,null,null,null,null,null,null,null,"p-value: Equivalent to alpha (type I error), once the test has been run"]
[null,"σ1 and σ2 : Standard deviation of each sample (necessary if they are different*).  See * in the \"Help\" worksheet )",null,null,null,null,null,null,null,null,null,null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products are similar"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"σ : Standard deviation of samples or an estimation of it"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"σ1 and σ2 : Standard deviation of each sample (necessary if they are different*).  See * in the \"Help\" worksheet )"]

{"sheet":"Thurstone","index":6,"n_rows":40,"n_cols":27,"rows_kept":40,"cut":false,"hidden":true,"merged":[[0,0,0,19],[17,18,17,25],[18,18,18,25],[19,18,19,25],[3,4,3,14],[7,5,7,6],[1,0,1,18],[4,4,4,14],[17,1,17,7],[12,9,12,16],[20,18,20,25],[11,1,11,7],[11,9,11,16],[12,1,12,7],[8,4,8,14]]}
["Unspecified Tetrads Test calculator"]
["Those results are effective in the case of N independent unspecified tetrads tests. If the tests include repetitions, check this paragraph :"]
[2]
[null,null,null,null,"Thurstonian approach"]
[null,null,null,null,"You are looking for the equivalence between δ (delta) and pD"]
[]
[null,null,null,null,null,"δ (delta)",null,null,null,null,null,null,"pD"]
[null,null,null,null,null,"       Use this value:"]
[]
[null,null,null,null,null,null,null,null,null,null,"                       For more information on Thurstonian models:"]
[]
[null,"BEFORE the test",null,null,null,null,null,null,null,"AFTER the test",null,null,null,null,null,null,null,null,null,null,"  INTERPRETATION"]
[null,"You are looking for the sample size",null,null,null,null,null,null,null,"You are looking for the parameters of the test",null,null,null,null,null,null,null," "]
[null,null,"δ (delta)",null,null,null,null,null,null,null,"N"]
[null,null,"α (alpha)",null,null,"N",null,null,null,null,"NC",null,null,null,"p-value"]
[null,null,"Power",null,null,null,null,null,null,null,"δ (delta)",null,null,null,"Power"]
[null,null,null,null,null,null,0.8,null,null,null,"α (alpha)"]
[null,"You are looking for the power",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null," "]
[null,null,"δ (delta)"]
[null,null,"α (alpha)",null,null,"Power"]
[null,null,"N"]
[]
[]
[null,null,"α (alpha) : type I error, risk of being wrong when saying the products are different",null,null,null,null,null,null,null,"α (alpha) : type I error, risk of being wrong when saying the products are different"]
[null,null,"δ (delta) : Distance between the two products, following the Thurstonian model",null,null,null,null,null,null,null,"δ (delta) : Distance between the two products, following the Thurstonian model"]
[null,null,"N : Sample size",null,null,null,null,null,null,null,"N : Sample size",null,null,null,null,null,null,null,null,null,null,null,null,"For further information:"]
[null,null,"pD : Proportion of distinguishers in the Guessing model, above which the products ",null,null,null,null,null,null,null,"NC : Number of people who had a correct answer (can be correct out of luck)"]
[null,null,"are considered different",null,null,null,null,null,null,null,"p-value : Equivalent to alpha (type I error), once the test has been run"]
[null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the",null,null,null,null,null,null,null,"Power : Power of the test, 1- type II error, chance of being right when saying the products are"]
[null,null,"products are similar",null,null,null,null,null,null,null,"similar"]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"D"]

{"sheet":"Help","index":7,"n_rows":279,"n_cols":1,"rows_kept":279,"cut":false,"hidden":true,"merged":[[263,0,264,0],[172,0,173,0]]}
["HELP"]
["For a printable version of this help, click here :"]
["Overall explanations"]
["What is power?"]
["Why use power?"]
["How does power increase?"]
["About replicates"]
[]
["Analyses details"]
["Discrimination tests"]
["Introduction"]
["Guessing model tests"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" tables"]
["Thurstonian model tests"]
["\"Thurstonian approach\" table"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
["Chi square tests"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
[]
["Comparisons of means"]
["Introduction"]
["One sample"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
["Two samples"]
["Dependent samples"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
["Independent samples"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
[]
["Overall explanations"]
["What is power?"]
["Essential definitions :"]
["All the tests used in this program are based on hypothesis testing. A null hypothesis, H0, is opposed to an alternative hypothesis, H1. The user has to choose one of them, considering two types of errors."]
[]
[]
[]
[]
[]
[]
["                                                                                                                                                                                                                       Figure 2 : Graphical representation of the errors of 1st kind (in red) and 2nd kind  "]
["                                                                 Figure 1 : Definitions of errors of 1st and 2nd kind                                                   (in blue). On this figure, we can see that if the result is in the red area, there is a"]
["                                                                                                                                                                                                                                                             low risk that it follows the distribution of H0."]
[]
["What happens is that the null hypothesis H0 is the hypothesis we start with, the one that is leading to a less problematic decision. H1 is the hypothesis that will be hard to prove. Indeed, the decision of rejecting H0 will only be taken if the result is much too far from what expected to be considered as pure chance. But if H1 is proved, then it is very probably true. Therefore, both errors are not on the same level : the error related to H0 will be higher than the one related to H1. That is why H0 should be the most harmless hypothesis."]
[]
["Let's consider the example of someone judged on a trial. The judge has to take a decision based on two hypothesis : the suspect is either innocent or guilty. The most harmless decision is to find the prevent innocent. Therefore, the hypothesis are :"]
["    H0 : \"The suspect is innocent.\""]
["    H1 : \"The suspect is guilty.\""]
["Therefore, there are two types of error, the first kind error α being the one we want to minimize :"]
["    α : Accuse an innocent."]
["    β : Set free a guilty person."]
[]
["Errors definitions :"]
["α : error of the first kind. Risk to falsely reject the null hypothesis H0 (i.e. probability to reject H0 when H0 is true).\nBefore the test, the user chooses an α which is the significance level. It is the probability below which the outcome of the test will be considered significant.\nAfter the test, the actual probability is called p-value or critical probability.\nIt is common to choose a very low α in order to minimize the risk of the first kind (α = 0.05 is a standard value in statistics)."]
["β : error of the second kind. Risk to falsely accept the null hypothesis H0 (i.e. probability to accept H0 when H0 is false).\nIt can either be calculated before the test or after the test, given the α value that determines the level of significance for the p-value.\nSince it is common to choose a very low α to minimize the risk of the first kind, the null hypothesis H0 will rarely be rightly rejected. It is then coherent that the risk to keep H0 when it is false will be higher. Indeed, the lower the α is, the higher β will be."]
["1-β : power of the test. Probability to rightly reject the null hypothesis H0 (i.e. to reject H0 when H0 is false).\nAs previously, it can be calculated before or after a test.\nIf the selected α is too low, the risk β is too high and hence the power of the test is too low."]
["To take again the example of the trial, we saw that the risk we minimize is to accuse an innocent Therefore a lot of times even guilty persons will be set free. The power of the test is the probability to accuse a guilty person. The higher it is, the more the test can be trusted."]
[]
["In sensory, the most common test is a difference test. Therefore, the hypothesis are :"]
["H0 : \"The products are identical.\""]
["H1 : \"The products are different\"."]
["considering :"]
["α : Risk to conclude the products are different when they are the same."]
["β : Risk to conclude the products are the same when they are different."]
["1-β (power) : Chance to conclude the products are different when they are."]
[]
["Why use power?"]
["   In the case of a difference test, when the result shows a significant difference between products (the p-value is under the significant level α), there is no doubt on the conclusion: the products are different. "]
["   However, the absence of significant difference between the products does not prove that the products are the same. In this situation, there are two possibilities : either the products are different, but the test was not strong enough to show it ; or the products are actually similar. No conclusion can be made."]
["   To prevent this situation, it is recommended to take a high power for the test. A high power means that when the conclusion is to keep the null hypothesis H0 (no significant difference), there is a high probability that it is correct : the products are similar. That way, by using a high power for the test, the user will always know which conclusion to make."]
[]
["   What V-Power does is to calculate the power and the required sample size before the test, so that the user can prevent this situation and always draw a conclusion whatever the test result is. It also provides the p-value and the power after the test in order to know if there is a difference or a similarity, with the associated risks."]
[]
["How does power increase?"]
["The higher the power is, the more the test is able to detect a similarity between products. The power increases with :"]
["- the actual difference between the products (effect size)"]
["- the variability of the samples (for example, the standard deviations of the groups in the case of a 2 samples t-test)"]
["- the sample size N"]
["- the value selected for α."]
["The last two criteria are the only ones the user can vary to increase the power of the test."]
[]
["About replicates"]
["Let’s consider the case of replicates in a sensory discrimination test where j judges perform each test k times."]
["The p-value is based on the hypothesis that “No distinction is possible between the products” by measuring how far we are from this hypothesis. It takes into account the statistical entities because one consumer would be enough to prove this hypothesis wrong. Therefore, the sample size to use to obtain the p-value is N = j x k."]
["However, the power depends on the number of judges who actually performed the test because it is based on the hypothesis that “Judges can see the difference between the products”. Therefore, you can only be proven wrong by considering more judges, which means that the entities are here the judges themselves. The proper sample size to compute power is then N = j."]
[]
["Analyses details"]
["Discrimination testing"]
["Introduction"]
["Discrimination tests are the tests practiced in sensory where consumers are asked to point out the product that is different from the others, in order to know if they can really perceive an overall difference between the products. The sensory discrimination tests provided by the V-Power program are :"]
["Classical sensory tests"]
["- The Triangle test : consumers are presented 3 products (one product X and 2 products Y). They must identify the single X product."]
["- The Duo-Trio test : consumers taste a reference product X, then are presented 2 products X and Y. They must identify the X product."]
["- The Two-out-of-Five test : consumers are presented 5 products (2 products X and 3 products Y). They must identify the two X products."]
["- The Same/Different test : consumers are presented 2 products, either X and X or X and Y. They must say if they are the same or different."]
["- The \"A\" - \"not A\" test : consumers are familiarized with both products \"A\" and \"not A\". Then the products are withdrawn and consumers are presented a sequence of products, \"A\" or \"not A\". They must identify the products one by one."]
["Alternative forced choice tests"]
["- The 2-AFC test : consumers are presented 2 products. They must identify the strongest on a specified attribute."]
["- The 3-AFC test : consumers are presented 3 products (one product X and 2 products Y). They must identify the strongest on a specified attribute."]
["- The Unspecified Tetrads test : consumers are presented 4 products (2 products X and 2 products Y). They must identify the 2 groups of products."]
["- The Specified Tetrads test : consumers are presented 4 products (2 products X and 2 products Y). They must identify the 2 strongest products on a specified attribute."]
[]
["To analyse discrimination tests, two models are available in the V-Power program : the Guessing model and the Thurstonian model."]
[]
["Guessing model"]
["The guessing model assumes that consumers are either distinguishers or non-distinguishers. Distinguishers always find the correct answer. Consumers being forced to answer, non-distinguishers are guessing the answer with a probability of a guess pG. Someone who does not taste a difference will still have 1 chance out of 2 to identify the good product for the duo-trio test, 1 out of 3 for the triangle test, etc. So with this model, the results are the same for the tests with a same pG : triangle, 3-AFC and Unspecified tetrads on one hand, and Duo Trio and 2-AFC on the other hand."]
["In this model, to determine if the products are different, the hypothesis relies on the estimation of the proportion of distinguishers pD."]
[]
["All the tests using the guessing model (Triangle, Duo trio and Two out of five) share the same tables structure."]
["\"BEFORE the test\" TABLES"]
["The first table calculates the minimum sample size N needed if you want the test to match the following parameters :"]
["- pD the proportion of distinguishers, the proportion of people who are able to actually detect a difference between the products. Here is defining the limit you want to fix to consider your products as different. For example, pD=0.20 means that you consider the products different from 20% of distinguishers."]
["- α (alpha) the risk of being wrong when saying the products are different."]
["- Power the chance of being right when saying the products are similar."]
[]
["The second table gives the Power of the test, given: "]
["- N the sample size"]
["- α (alpha)"]
["- pD the proportion of distinguishers, the proportion of people who are able to actually detect a difference between the products. Here is defining the limit you want to fix to consider your products as different. For example, pD = 0.20 means that you consider the products different from at least 20% of distinguishers."]
[]
["\"AFTER the test\" TABLES"]
["The first table calculates the proportion of distinguishers pD of the test and to build a confidence interval around this value, given :"]
["- N the sample size"]
["- NC the number of correct answers, which is different from people who actually distinguished the products. Indeed, the respondents have to give an answer, (which can be correct) even if they do not taste the difference."]
["- α of confidence interval the parameter you choose to build the confidence interval. For example, α=0.1 means that there is a probability of 90% that pD is between the minimum and maximum of the interval. This means that there is 5% error above the upper limit and 5% below the lower limit."]
["Therefore, if you are only interested in the upper limit (if you want to prove a similarity for example) then you can be assured of a 95% confidence to be below the maximum of the interval."]
[]
["The second table gives the p-value and the Power of the test given :"]
["- N the sample size"]
["- NC the number of correct answers, which is different from people who actually distinguished the products. Indeed, the respondents have to give an answer, (which can be correct) even if they do not taste the difference."]
["- pD the proportion of distinguishers, the proportion of people who are able to actually detect a difference between the products. Here is defining the limit you want to fix to consider your products as different. For example, pD = 0.20 means that you consider the products different from at least 20% of distinguishers."]
[]
["Thurstonian model"]
["With this model, the required parameter is not a proportion of distinguishers pD but δ (delta), the sensory distance between the two products, where one unit represents a standard deviation. The assumptions are that the sensory representations of the products follow normal distributions and that the consumers are not categorized as distinguishers/non-distinguishers. Consumers are always correct, translating what they perceive. Thus an incorrect answer corresponds to a proximity between products that leads to an incorrect perception (figure 3)."]
[]
[]
[]
[]
["Figure 3 : Thurstonian model for Triangle test : the Y product has to be detected. The smaller δ is, the more often the products will be misclassified"]
[]
["All the discrimination tests treated with thurstonian modelling are based on the same calculations. The differences are : "]
["- the probability of a guess pG (correct answers by chance, since consumers are forced to make a choice). Someone who does not taste a difference will still have 1 chance out of 2 to identify the correct product for the 2-AFC and Duo-Trio test, 1 out of 3 for the 3-AFC, Triangle and Unspecified Tetrads test and 1 out of 6 for the Specified Tetrads test."]
["- the Psychometric function, which links delta to pC the proportion of correct answers, is different for every test."]
["In this model, to determine if the products are different, the hypothesis relies on the estimation of the sensory distance between the products δ."]
[]
["All the tests using the thurstonian model (Triangle, Duo Trio, 2-AFC, 3-AFC and Tetrads) share the same tables structure."]
["\"Thurstonian approach\" TABLE"]
["The first table creates a link between the guessing model and the thurstonian models. It enables to go back and forth between:"]
["- pD the proportion of distinguishers from the guessing model"]
["- δ (delta) the distance between the products in the thurstonian model"]
["Those two models are incompatible and δ would be the best parameter to use. But because pD is a more understandable and a more common notion than δ, this table is a way to obtain equivalences. For example, a proportion of distinguishers pD = 20% in a triangle test is equivalent to a distance δ = 1,28 between products.\nThe proportion of correct answers pC is the link between the two parameters because it is used in both models."]
[]
["\"BEFORE the test\" TABLES"]
["The first table calculates the minimum sample size N needed if you want the test to match the following parameters:"]
["- δ (delta)"]
["- α (alpha) the risk of being wrong when saying the products are different"]
["- Power the chance of being right when saying the products are similar"]
[]
["The second table gives the Power of the test, given:"]
["- N the sample size"]
["- δ the distance between the products"]
["- α (alpha)"]
[]
["\"AFTER the test\" TABLE"]
["The table calculates the proportion of distinguishers pD and to build a confidence interval around this value, given :"]
["- N the sample size"]
["- NC the number of people who answered correctly, which is different from people who actually distinguished the products. Indeed, the respondents have to give an answer (which can be true) even if they do not taste a difference."]
[]
["- δ (delta)"]
["- α (alpha)"]
[]
["Chi square tests"]
["The Same/Different and the \"A\"-\"not A\" tests share a statistical treatment very different from the other discrimination tests. "]
["Here, the data are organized in a 2 by 2 contingency table, crossing \"Same\" and \"Different\" products (respectively \"A\" and \"not A\" products) with the answers \"Same\" and \"Different\" for consumers (respectively \"A\" and \"not A\" for consumers)."]
["In this model, to determine if the products are different, the hypothesis relies on the repartition into the contingency table (allotments). The test will compare the allotments obtained with consumers to the allotments that would be due only to hazard (i.e. by multiplying the margins values 2 by 2). "]
[]
["The statistical distribution that allow to compare those allotments is called the Chi-square (χ2) distribution."]
[]
["\"BEFORE the test\" TABLES"]
["The first calculates the minimum sample size N needed if you want the test match the following parameters :"]
["- φ (phi) the effect size : it gives a measure of the strength of the relationship between the two variables. Since the test has not been run yet, value references are written below. 0.1 is small, 0.3 is average and 0.5 is a strong effect"]
["- α (alpha) the risk of being wrong when saying the products are different"]
["- Power the chance of being right when saying the products are similar"]
[]
["The second table gives the power of the test, given:"]
["- N the sample size"]
["- α (alpha)"]
["- ω the effect size"]
[]
["\"AFTER the test\" TABLE"]
["The table of the test provides 5 results:"]
["- The effect size phi (φ) calculated with the data."]
["- χ2 (Chi square), the test statistic value. This value is the one that will be used for the test parameters calculation."]
["- p-value : if this one is significant (at the threshold fixed by the user), it means that the allotments are too far away from a random allotment. It can mean two different things, depending on how the allotment is."]
["    - If the diagonal of the contingency table owns the highest numbers, it means that consumers saw the difference between products, they recognized more than their normal when products where actually different."]
["    - If the diagonal of the contingency table owns the lowest numbers, it means that consumers didn't see the difference between products. Indeed, a significance in that way shows that consumers responded more often that products were different when they were the same and more often they where identical when they were different, which does not make sense."]
["- λ is the non centrality parameter. It is necessary to calculate the power of the test, since it is based on this value that is built the non-central χ2 distribution."]
["- Power is the test power. It is using the p-value result, but also the non-central parameter λ calculated thanks to the α value given by the user."]
[]
["Comparisons of means"]
["Introduction"]
["  Those tests are statistical comparisons of means. They are testing the difference based on a specific attribute that will have a different response for each product. For example, a firm might want to check several parameters of their food process, such as the amount of an ingredient, the weight of the final product... Or the overall liking between two types of their products.\n  They can either test the difference between two samples on their means or a single sample and compare it to a theoretical mean (). It is useful when you want to know if products are leading to the same response for an attribute, for example."]
["  Two questions are asked before each calculation in order to define the situation."]
["1. \"Is it a one or a two-sided test? (i.e. are you interested in a superiority/ inferiority or just in a difference)\"\nThis question will precise the test conditions :\n- Two-sided test : If you are looking for a difference between two means, whether it is a lower or a higher mean, then a two-sided test is necessary.\n- One-sided test : If you are looking for a superiority or inferiority only, then a one-sided test is necessary. The a priori knowledge of your data (\"This amount of sugar can never go lower than this theoretical value so I don't need to look at this side\"), or the more freedom you give to your judgment on the results (\"It is fine if the amount of milk in each bottle is superior to 1 liter, I just want to be able to detect if the amount is lower than 1 liter\") is leading to stronger results for one-sided test. For example, less people are needed to demonstrate a superiority of a mean than to demonstrate a simple difference."]
["2. \"Is the standard deviation an estimation?\"\nThis question will define the type of distribution that is used to run the test. Indeed, there are two possible distributions :\n- The Normal distribution : If the standard deviation of the sample(s) is considered as the absolute one, then this distribution is used. This is the most common distribution employed for this test in sensory programs.\n- The Student distribution : If the standard deviation of the sample(s) is the one calculated on the sample, or an estimation, then the distribution for the test needs to be the student distribution. Indeed, by calculating a parameter with the sample(s) used for the test, the distribution is slightly different and will lead to weaker results (for example, the power will be slightly lower with the Student distribution for the same test dataset). The statistical reason for it is that a degree of freedom is used to calculate this parameter and hence there is less strength for the test itself."]
["By default, tests are considered as two-sided and using the Normal distribution. It is the case if the user does not give the information (cancels the windows)."]
[]
["One sample"]
["\"BEFORE the test TABLES\""]
["The first table allows to calculate the minimum sample size N needed if you want the test to have sufficiently good parameters, which are:"]
["- α (alpha) critical : threshold for the type I error, risk of being wrong when rejecting the null hypothesis (i.e. probability to be wrong by considering that the product mean is the same than the theoretical value)."]
["- 1-β (1-beta, or power) : threshold for 1- type II error, chance of being right when accepting the alternative hypothesis (i.e. probability of being right by considering that the product mean is different than the theoretical value)."]
["- Delta (δ) : Minimum difference between the product mean and the theoretical value that you want to be able to notice (i.e. the value above which the product is considered too different)."]
["- Standard deviation (σ) : Either the theoretical standard deviation (it can be established that a certain type of machine has a given standard deviation) or the sample standard deviation (an estimation of the real standard deviation). The question is asked further which of those two possibilities is used."]
[]
["The second table gives 1-β the power of the test, given the sample size you will have, the threshold you want to fix for α (alpha) and δ (delta)."]
[]
["\"AFTER the test\" TABLE"]
["The table gives the parameters α (p-value) and 1-β (power) of the test given :"]
["- N : Actual sample size you had for the test"]
["- Delta measured (δm) : Actual difference between the product mean and the theoretical value."]
["- Standard deviation (σ) : Either the theoretical standard deviation (it can be established that this type of machine has a given deviation) or the sample standard deviation (an estimation of the real standard deviation). The question is asked further which of those two possibilities is used."]
["- Delta for power (δ) : Minimum difference between the products you want to be able to notice (i.e. the value above which the products are considered as different). This value is useful for the alternative hypothesis in order to calculate the test power."]
[]
["Two samples"]
["Dependent samples  / Paired sample"]
["This method is used when the samples are correlated. It is the case if products are given as a pair, or in the case of 'before/after' studies, when the interest is to check if there is an evolution, an effect of a specific treatment or an effect of time on consumers perceptions."]
["For this kind of test, it is assumed that the theoretical standard deviations σ1 and σ2 of the two samples are the same. Therefore, no previous test is run to check if the two standard deviations are identical or not."]
[]
["\"BEFORE the test\" TABLES"]
["The first table allows to calculate the minimum sample size N needed if you want the test to have sufficiently good parameters, which are:"]
["- α (alpha) critical : significance level for the type I error, risk of being wrong when rejecting the null hypothesis (i.e. probability to be wrong by considering that the products are the same)."]
["- 1-β (1-beta, or power) : 1- type II error, chance of being right when accepting the alternative hypothesis (i.e. probability of being right by considering that the products are different)."]
["- Delta (δ) : Minimum difference between the products you want to be able to notice (i.e. the value above which the products are considered as different)."]
["- Standard deviation (σ) : Either the theoretical standard deviations (it can be established that a certain type of machine has a given standard deviation) or the samples standard deviation (estimations of the real standard deviations). The question is asked further which of those two possibilities is used."]
[]
["The second table gives the power of the test, given the sample size you will have, the threshold you want to fix for α (alpha) and δ (delta)."]
[]
["\"AFTER the test\" TABLE"]
["The table gives the parameters α (p-value) and 1-β (power) of the test given :"]
["- N : Actual sample size (pairs of observation) you had for the test"]
["- Delta measured (δm) : Actual difference between the two groups means"]
["- Standard deviation (σ) : Either the theoretical standard deviation (it can be established that this type of machine has a given deviation) or the sample standard deviation (an estimation of the real standard deviation)."]
["- Delta for power (δ) : Minimum difference between the means you want to be able to notice (i.e. the value above which the two means are considered as different). This value is useful for the alternative hypothesis in order to calculate the test power."]
[]
[]
["Independent samples"]
["\"BEFORE the test\" TABLES"]
["The first table allows to calculate the minimum sample size N needed if you want the test to have sufficiently good parameters, which are:"]
["- α (alpha) critical : threshold for the type I error, risk of being wrong when rejecting the null hypothesis (i.e. probability to be wrong by considering that the products are the same)."]
["- 1-β (1-beta, or power) : threshold 1- type II error, chance of being right when accepting the alternative hypothesis (i.e. probability of being right by considering that the products are different)."]
["- Delta (δ) : Minimum difference between the products you want to be able to notice (i.e. the value above which the products are considered as different)."]
["- Standard deviation (σ) : Either the theoretical standard deviations (it can be established that a certain type of machine has a given standard deviation) or the samples standard deviation (estimations of the real standard deviations). The question is asked further which of those two possibilities is used."]
[]
["The second table gives the Power of the test, given the sample size you will have, the threshold you want to fix for α (alpha) and δ (delta)."]
[]
["\"AFTER the test\" TABLE"]
["Before comparing the samples means through a t-test, the samples standard deviations have to be compared using a F-test. The results of this first test will influence the comparison of means. Indeed, whether the standard deviations are similar or not will give different parameters for the t-test."]
[]
["The F test is also a hypothesis test, with :\n   H0 the null hypothesis (σ1 = σ2)\n   H1 the alternative hypothesis (σ1 ≠ σ2)"]
["The test statistic is F= S1/S2, fraction of the two approximated standard deviations S1 and S2. The distribution used for this test is the Fisher distribution, giving its name to the test."]
[" .  If the test does not give a significant difference between the standard deviations, the parameters are :"]
["   S' (global standard deviation) = sqrt( ((n1-1)S12+(n2-1)S22)/(n1+n2-2) )* sqrt(1/n1+1/n2)    "]
["   df (degrees of freedom) = (n1 -1) + (n2 -1)"]
[" .  If the test gives significant difference at 5%, the user can still choose to consider them as identical. If not, the new parameters are :"]
["   S' (global standard deviation) = sqrt( S12/n1 + S22/n2 )"]
["   df = (S12/n1 + S22/n2)2 / ( (S12/n1)2/(n1-1)+(S22/n2)2/(n2-1) )"]
[]
["The table gives the parameters α (p-value) and 1-β (power) of the test, given :"]
["- N : Actual sample size you had for the test"]
["- Delta measured (δm) : Actual difference between the products means."]
["- Standard deviation (σ) : Either the theoretical standard deviations (it can be established that this type of machine has a given deviation) or the samples standard deviations (estimations of the real standard deviations). The question is asked further which of those two possibilities is used."]
["- Delta for power (δ) : Minimum difference between the products you want to be able to notice (i.e. the value above which the products are considered as different). This value is useful for the alternative hypothesis in order to calculate the test power."]

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["CALCULATION"]
[]
["Guessing model tests"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" tables"]
[]
["Thurstonian model tests"]
["\"Thurstonian approach\" table"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
[]
["Chi² distribution tests"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
[]
["Comparisons of means"]
["One Sample"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
["Two Samples"]
["Dependent /Paired Samples"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
["Independent Samples"]
["\"BEFORE the test\" tables"]
["\"AFTER the test\" table"]
[]
[]
["Guessing model tests"]
[]
["The program is using the Excel \"Binomial\" formula : Binomial ( NC = Number of correct answers, N = Number of answers, P = Probability of success, Cumulative = True)"]
["It returns the Probability that a Binomial distribution of parameters P and N obtains a number of success inferior or equal to NC."]
["The reverse distribution of the Binomial distribution is Criterebinom ( N = Number of answers, P = Probability of success, Alpha = Criteria for the cumulative probability of the Binomial distribution)"]
["It returns the minimum NC for which the Binomial distribution is superior or equal to alpha. What Excel is calling alpha is actually (1 - the statistical alpha)."]
[]
["\"BEFORE the test\" TABLES"]
["N: Loop calculating N one by one (starting from N=3), following the formulas of power until"]
["power ≥ value fixed by user"]
[null,"where N is the sample size"]
["power = Binomial ( Criterebinom(N, P, 1-αgiven by user), N, P= pC for pD )"]
[]
["\"AFTER the test\" TABLE"]
["pD = (pC - pG) / (1 - pG)"]
[null,"where pD is the proportion of distinguishers"]
[null,"where pC is the proportion of correct answers"]
[null,"where pG is the probability of a guess : pGTriangle = 1/3,  pGDuoTrio = 1/2 and  pGTwoOutOfFive=1/10"]
["CI = p ± q*σ"]
[null,"where CI is the confidence interval "]
[null,"where q is the quantile of the Normal distribution for 1-α/2"]
[null,"where σ is the standard deviation : σ = x √( pC*(1-pC)/N )   (xTriangle=3/2, xDuoTrio = 2 and xTwoOutOfFive= 10/9)"]
["p-value = 1- Binomial (Nc, N, pG)"]
[null,"where NC is the number of correct answers"]
[null,"where N is the sample size"]
[null,"where pG is the probability of a guess : pGTriangle = 1/3, pGDuoTrio = 1/2 and pG2outof5 = 1/10"]
["power = Binomial ( Criterebinom(N, pG, 1-α), N, pC )"]
[null,"where N is the sample size"]
[null,"where pG is the probability of a guess : pGTriangle = 1/3, pGDuoTrio = 1/2 and pG2outof5 = 1/10"]
[null,"where pC is the proportion of correct answers : pC = pD + pG (1- pD)"]
[]
["Thurstonian model tests"]
[]
["The program is using the Excel \"Standardized Normal\" distribution NormalS( x = value to compare to the standardized Normal distribution)"]
["It returns the cumulative probability in x for the Normal distribution of parameters mu = 0, sd = 1."]
["The reverse distribution of the Standardized Normal distribution is InvNormalS( p = cumulated probability of the standardized Normal distribution)"]
["It returns the value that would give a cumulated probability of p for the standardized Normal distribution."]
["The program is using the integral from the New Functions for Excel(2) Integral(\"formula\", a = Minimum, b = Maximum, l = Level of precision, False=Boolean to show the result in a window)"]
["It calculates the integral for the formula between two numbers a and b, at the level of precision l (number of loops to obtain the results)"]
[]
["\"Thurstonian approach\" TABLE"]
["pD = (pC - pG) / (1 - pG)"]
[null,"where pD is the proportion of distinguishers"]
[null,"where pC is the proportion of correct answers calculated from δ (delta) using :"]
[null,"The Psychometric functions"]
[null,"     For 2-AFC :"]
[null,"pC = NormalS( δ /√2 )\""]
[null,"     For 3-AFC :"]
[null,"pC = integral(\"\"1/√(2pi)*NormalS²(x)*exp(-(x-δ)²/2)\"\", -50, 50, 3, False)"]
[null,"     For Unspecified Tetrads :"]
[null,"pC = 1-2/√(2pi)*integral(\"\"exp(-x²/2)* (2*NormalS(x)*NormalS(x-δ)- [NormalS(x-δ)]²)\"\", -50, 50, 3, False)\""]
[null,"     For Specified Tetrads :"]
[null,"pC = 1-2/√(2pi)*integral(\"\"exp(-x²/2)*NormalS(x)*(2*NormalS(x-δ)-[NormalS(x-δ)]²)\"\", -50, 50, 3, False)"]
[null,"     For Triangle :"]
[null,"pC =2/√(2pi)*integral(\"\"exp(-x²/2)*(NormalS(-x*√3 + δ*√(2/3) ) + NormalS(-x*√3 - δ*√(2/3) ))\"\", 0, 50, 3, False)\""]
[null,"     For Duo-Trio :"]
[null,"pC = 1- NormalS( δ /√2 )- NormalS( δ/√6 )+ 2*NormalS( δ/√2 )*NormalS( δ/√6 )"]
[null,"where δ is the distance (in standard deviations) between the two products"]
[null,"where pG is the probability of a guess : pG3-AFC, Unspecified Tetrads and Triangle = 1/3,  pG2-AFC and Duo Trio = 1/2 and  pGSpecified Tetrads = 1/6"]
[]
["δ (delta) : First pC is calculated from pD by the formula pC = pD + pG (1- pD)"]
["Then a dichotomic calculation is used, recalculating pC with the proper psychometric function at each step."]
[null,"By taking delta between 0 and 10 as a first length, and considering 11 steps for the dichotomy, this leads to an error <0.0025 (=5/211)"]
[]
["\"BEFORE the test\" TABLES"]
["N : Loop calculating N one by one (starting from N=3), following the proper formula of power until"]
["power ≥ value fixed by user"]
[null,"where N is the sample size"]
["Normal distribution"]
["Power = 1 - InvNormal ( pCalpha, pC, sd2)"]
[null,"where pCalpha = InvNormal (1-alpha, pG, sd1)"]
[null,"where sd1 is the standard deviation under the H0 hypothesis : sd1= √( pG*(1-pG)/N )"]
[null,"where pC is the proportion of correct answer (calculated from delta by the psychometric function)"]
[null,"where sd2 is the standard deviation under the H1 hypothesis : sd2 = √( pC*(1-pC)/N )"]
["Binomial distribution"]
["Power = 1 - Binomial ( Criterebinom(N, pG, α), N, pC, True )"]
[null,"where N is the sample size"]
[null,"where pG is the probability of a guess : pG3-AFC, Unspecified Tetrads and Triangle = 1/3,  pG2-AFC and Duo Trio = 1/2 and  pGSpecified Tetrads = 1/6"]
[null,"where α is the type I error : level of significance for the p-value"]
[null,"where pC is the proportion of correct answer (calculated from delta by the psychometric function)"]
[]
["\"AFTER the test\" TABLE"]
["Normal distribution"]
["α (p-value) = 1 - Normal (pC, pG, sd1)"]
[null,"where pC is the actual proportion of correct responses in the test : pC = NC/N"]
[null,"where pG is the probability of a guess : pG3-AFC, Unspecified Tetrads and Triangle = 1/3,  pG2-AFC and Duo Trio = 1/2 and  pGSpecified Tetrads = 1/6"]
[null,"where sd1 is the standard deviation under the H0 hypothesis : sd1= √( pG*(1-pG)/N )"]
["Power = 1 - Normal (pCalpha, pC, sd2)"]
[null,"where pCalpha = InvNormal (1-alpha, pG, sd1)"]
[null,"where pC is the proportion of correct answer (calculated from delta by the psychometric function)"]
[null,"where sd2 is the standard deviation under the H1 hypothesis : sd2 = √( pC*(1-pC)/N )"]
["Binomial distribution"]
["α (p-value) = 1 - Binomial (NC, N, pG, True)"]
[null,"where NC is the number of correct answers"]
[null,"where N is the sample size"]
[null,"where pG is the probability of a guess : pG3-AFC, Unspecified Tetrads and Triangle = 1/3,  pG2-AFC and Duo Trio = 1/2 and  pGSpecified Tetrads = 1/6"]
["Power = 1 - Binomial ( Criterebinom(N, pG, α), N, pC, True )"]
[null,"where N is the sample size"]
[null,"where pG is the probability of a guess : pG3-AFC, Unspecified Tetrads and Triangle = 1/3,  pG2-AFC and Duo Trio = 1/2 and  pGSpecified Tetrads = 1/6"]
[null,"where α is the type I error : level of significance for the p-value"]
[null,"where pC is the proportion of correct answer (calculated from delta by the psychometric function)"]
[]
["Chi2 distributions tests"]
[]
["The Same/Different test and the \"A\"-\"Not A\" test are sharing the same calculation. The results are exacts for the Same/Different test, and approximations for the \"A\"-\"Not A\" test."]
[]
["The program is using the Excel \"ChiSquare\" formula : ChiSquare( x = value to compare the Chisquare distribution to , df(degrees of freedom) =1 )"]
["It returns the Probability that a χ2 (Chi2) distribution with 1 degree of freedom obtains a value higher than x."]
["The program is using the Excel \"ChiSquareInv\" formula : ChiSquareInv( p = probability of the Chisquare distribution , df(degrees of freedom) =1 )"]
["It returns the value x above which the χ2 distribution with 1 degree of freedom obtains a probability lower than p."]
["The non-central χ2 distribution is necessary to calculate the power. Since it does not exist in Excel, the Sankaran(1) approximation is used. It is based on the Normal distribution."]
[]
["\"BEFORE the test\" TABLES"]
["N : Loop calculating N one by one (starting from N=3), following the formulas of alpha and beta with power (1-β) ≥ value fixed by user"]
["power = Sankaran's approximation below"]
[null,"where x = N* φ2"]
[]
["\"AFTER the test\" TABLE"]
["χ2 (Chi square) = Σfor the 4 cells of the table ( Obs-Th )2/ Th"]
[null,"where Obs = Observed number"]
[null,"where Th = Theoretical number"]
["α (alpha) = ChiSquare( χ2, 1 )"]
[null,"where it is always 1 degree of freedom because the table has 4 cells"]
["power = NormalS[ (1 - hp (1-h+ 0.5(2-h)mp) - (x / (1+λ)h) ) / ( h√(2p(1 + mp)) ) ]*"]
[null,"where h = 1- 2/3( ((k+λ)(k+3λ)) / (k+2λ)2 )"]
[null,"where p = (k+2λ)/(k+λ)2"]
[null,"where m = (h-1)(1-3h)"]
[null,"where λ is the non-centrality parameter : λ = ChiSquareInv(α,1)",null,null,null,null,null," "]
[null,"where x is the value to compare the non central χ2 distribution to"]
[" *Sankaran's approximation(1)"]
[]
[]
["Comparisons of means"]
[]
["The program is using the Excel \"Standardized Normal\" distribution NormalS( x = value to compare to the standardized Normal distribution)"]
["It returns the cumulative probability in x for the Normal distribution of parameters mu = 0, sd = 1."]
["The reverse distribution of the Standardized Normal distribution is InvNormalS( p = cumulated probability of the standardized Normal distribution)"]
["It returns the value that would give a cumulated probability of p for the standardized Normal distribution."]
["The program is also using the Excel \"Student\" distribution Student( x = value to compare to the Student distribution, df = degrees of freedom, one-sided)"]
["It returns the tail of a Student distribution with df degrees of freedom."]
["The reverse distribution of the Student distribution is InvStudent( p= Probability of the tail distribution for the two-sided Student distribution, df = degrees of freedom)"]
["It returns the value that would give a probability p/2 for the tail of a Student distribution with df degrees of freedom."]
[]
["ONE SAMPLE"]
["z ( uobs or tobs depending on the distribution used ) = Dm / (σ/ √(N) )"]
[]
["\"BEFORE the test\" TABLES"]
["Normal distribution"]
["     One-sided test : "]
["N = (σ² /δ² ) * ( Invnorm(1- α) - Invnorm(power) )²"]
["power = NormalS( InvNormalS(1-α) - D/(σ/√(N)) )"]
["     Two-sided test : "]
["N = (σ2 /δ2) * ( Invnorm(1- α/2) - Invnorm(power) )2"]
["power = NormalS( InvNormalS(α/2) - D/(σ/√(N)) ) + NormalS( InvNormalS(1-α/2) - D/(σ/√(N)) )"]
[]
["Student distribution"]
["N: Loop calculating N one by one (starting from N=3), following the power formula below until"]
["power ≥ value fixed by user"]
["One-sided test : power = Student( Invstudent( α, N-1) - D/( σ/√(N) ), N-1)"]
["Two-sided test : power = Student( -Invstudent( α/2, N-1) - D/( σ/√(N) ), N-1 ) + Student( Invstudent( α/2, N-1) - D/( σ/√(N)) , N-1 )"]
[]
["\"AFTER the test\" TABLE"]
["Normal distribution"]
["     One-sided test : "]
["p-value = 1 - NormalS(z)"]
["power = NormalS( InvNormalS(1-α) - D/(σ/√(N)) )"]
["     Two-sided test : "]
["p-value = 2 *( 1 -NormalS(z) )"]
["power = NormalS( InvNormalS(α/2) - D/(σ/√(N)) ) + NormalS( InvNormalS(α/2) - D/(σ/√(N)) )"]
[]
["Student distribution : the standard deviation is an approximation"]
["     One-sided test : "]
["p-value = Student(z, N-1)"]
["power = Student( Invstudent(α, N-1) - D/( σ/√(N) ), N-1)"]
["     Two-sided test : "]
["p-value = 2*Student(z, N-1)"]
["power = Student( -Invstudent( α/2, N-1) - D/( σ/√(N) ), N-1 ) + Student( Invstudent( α/2, N-1) - D/( σ/√(N)) , N-1 )"]
[]
[null,"where N is the sample size"]
[null,"where σ is the standard deviation"]
[null,"where Dm is the actual difference between the sample mean and the theoretical value"]
[null,"where D is the threshold above which the means are considered different"]
[null,"where α is the type I error : level of significance for the p-value"]
[null,"where z is the test statistic"]
[]
["TWO SAMPLES"]
["Dependent samples  / Paired samples"]
["For dependent samples:"]
["   the parameters are calculated with the Student distribution"]
["   df = N-1"]
["   t = D * √(N) /(√(s12+ s22- 2*r*s1*s2)"]
[null,"where df is the degrees of freedom"]
[null,"where N is the number of pairs"]
[null,"where t is the test statistic"]
[null,"where D is the mean of the differences (or the difference of the means)"]
[null,"where s1 and s2 are the standard deviations of the two samples"]
[null,"where r is the correlation coefficient between the two samples"]
[]
["\"BEFORE the test\" TABLES"]
[]
["N: Loop calculating N one by one (starting from N=3), following the formula below and with:"]
["power ≥ value fixed by user"]
["One-sided test : power = Student( Invstudent(2*α,N-1) - t, N-1)"]
["Two-sided test : power = Student( -Invstudent(α,N-1) - t, N-1) + Student( Invstudent(α,N-1) - t, N-1)"]
[]
["\"AFTER the test\" TABLE"]
["     One-sided test : "]
["p-value = Student(t, N-1)"]
["power = Student( Invstudent(2*α,N-1) - t, N-1)"]
["     Two-sided test : "]
["p-value = 2*Student(t, N-1)"]
["power = Student( -Invstudent(α,N-1) - t, N-1) + Student( Invstudent(α,N-1) - t, N-1)"]
[]
["Independent samples"]
["A F-test is run to compare the samples standard deviations"]
["If the standard deviations are not significantly different at 5%"]
["     S' = √( ((n1-1)S12+(n2-1)S22)/(n1+n2-2) ) * √(1/n1+1/n2)    "]
["     df = n1 + n2 - 2"]
["If the standard deviations are significantly different at 5% (then the user is still able to consider them as identical)"]
["     S' = √( S12/n1 + S22/n2 )                                                                            "]
["     df = (S12/n1 + S22/n2)2 / ( (S12/n1)2/(n1-1)+(S22/n2)2/(n2-1) )"]
[]
["z = (D or Dm)/ S'"]
["     If n1 = n2 = N/2,   z = Dm/ √( 2/N*(S12+S22) )"]
["     If S1 = S2 = S,     z = Dm/ √( S2*(1/n1+1/n2) )"]
[]
[null,"where S' is the global standard deviation"]
[null,"where S1 and S2 are the standard deviations of the samples"]
[null,"where D is the difference above which the means are considered different"]
[null,"where Dm is the difference mesured between the means"]
[null,"where N is the total size"]
[null,"where n1 and n2 are the samples sizes"]
[null,"where df are the degrees of freedom"]
[null,"where z is the test statistic"]
[]
["\"BEFORE the test\" TABLES"]
["Normal distribution"]
["     One-sided test : "]
["N = 2*((S12+S22)/ D2)*( Invnorm(1-α) - Invnorm(1-β) )2"]
["power = NormalS( InvNormalS(α) - z )"]
["     Two-sided test : "]
["N = 2*((S12+S22)/ D2)*( Invnorm(1-α/2) - Invnorm(1-β) )2"]
["power = NormalS( -InvNormalS(α/2) - z ) + NormalS( InvNormalS(α/2) - z )"]
[]
["Student distribution"]
["N: Loop calculating N one by one (starting from N=3), following the power formula below until"]
["power ≥ fixed value for power"]
["One-sided test : power = Student( Invstudent(2*α,df) - z, df)"]
["Two-sided test : power = Student( -Invstudent(α,df) - z, df) + Student( Invstudent(α,df) - z, df)"]
[]
["\"AFTER the test\" TABLE"]
["Normal distribution"]
["     One-sided test : "]
["p-value = 1 - NormalS(z)"]
["power = NormalS( InvNormalS(α) - z )"]
["     Two-sided test : "]
["p-value = 2*( 1 - NormalS(z) )"]
["power = NormalS( -InvNormalS(α/2) - z ) + NormalS( InvNormalS(α/2) - z )"]
[]
["Student distribution"]
["     One-sided test : "]
["p-value = Student(z, df)"]
["power = Student( Invstudent(2*α, df) - z, df)"]
["     Two-sided test : "]
["p-value = 2* Student(z, df)"]
["power = Student( -Invstudent(α,df) - z, df) + Student( Invstudent(α,df) - z, df)"]
[]
[]
["(1) Sankaran, M. (1963). Approximation to the non-central chi-square distribution Biometrika 1963 50(1-2):199-204"]
["(2) New Functions for Excel are available at www.fordom.free.fr"]