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Contingency Tables
Contingency Tables
Published: 2013/04/03
Channel: Ciarán Mac an Bhaird
Contingency table chi-square test | Probability and Statistics | Khan Academy
Contingency table chi-square test | Probability and Statistics | Khan Academy
Published: 2010/11/11
Channel: Khan Academy
6. Probabilities Contingency Table
6. Probabilities Contingency Table
Published: 2010/08/20
Channel: Red River College - Tutoring
contingency table intro
contingency table intro
Published: 2011/10/02
Channel: MrNystrom
Contingency Table Basics
Contingency Table Basics
Published: 2015/05/03
Channel: Stephanie Glen
Chi Square Test - with contingency table
Chi Square Test - with contingency table
Published: 2015/05/19
Channel: Math Meeting
Using Contingency Tables
Using Contingency Tables
Published: 2014/06/26
Channel: Mindset Learn
Probability Example: Conditional Probability with a Contingency Table
Probability Example: Conditional Probability with a Contingency Table
Published: 2015/12/28
Channel: Josiah Hartley
Week 7 : TUTORIAL: CONTINGENCY TABLES
Week 7 : TUTORIAL: CONTINGENCY TABLES
Published: 2012/12/17
Channel: Stata Learner
Contingency Tables
Contingency Tables
Published: 2015/05/01
Channel: MaestasMath
Contingency Tables and Joint Probability Tables
Contingency Tables and Joint Probability Tables
Published: 2012/10/02
Channel: sturmmath
Contingency Table - Total Percentage
Contingency Table - Total Percentage
Published: 2012/07/06
Channel: fuquanda
INDEPENDENT OR ASSOCIATION? Reading a CONTINGENCY TABLE
INDEPENDENT OR ASSOCIATION? Reading a CONTINGENCY TABLE
Published: 2011/10/02
Channel: MrNystrom
Probabilities from a Contingency Table/Joint Probability Table
Probabilities from a Contingency Table/Joint Probability Table
Published: 2016/02/07
Channel: Joshua Emmanuel
Elementary Statistics: Contingency Tables and Association
Elementary Statistics: Contingency Tables and Association
Published: 2012/09/23
Channel: David Hays
contingency tables
contingency tables
Published: 2008/10/08
Channel: headlessprofessor
Calculate Contingency Table and Expected Values for Chi-squared Test with Excel
Calculate Contingency Table and Expected Values for Chi-squared Test with Excel
Published: 2013/11/17
Channel: Temenoujka Fuller
Test of Independence (Contingency Tables)
Test of Independence (Contingency Tables)
Published: 2013/11/10
Channel: Kelly Sjerven
Creating Contingency Tables in Excel
Creating Contingency Tables in Excel
Published: 2015/06/01
Channel: Erik Heineman
Understanding Contingency Tables and Crosstabulation, Pt. 1
Understanding Contingency Tables and Crosstabulation, Pt. 1
Published: 2013/08/22
Channel: Alan Neustadtl
Probability intro to contingency tables lesson
Probability intro to contingency tables lesson
Published: 2009/10/21
Channel: TwoMinuteTeacher
2 by 2 Contingency Table Analysis (Pearson Chi-Square) - SPSS (part 1)
2 by 2 Contingency Table Analysis (Pearson Chi-Square) - SPSS (part 1)
Published: 2011/09/20
Channel: how2stats
Contingency Tables on the Ti83 Calculator
Contingency Tables on the Ti83 Calculator
Published: 2012/12/02
Channel: daveahlberg123
contingency table
contingency table
Published: 2013/04/02
Channel: Tom Sutton
Contingency Table -CountIF - Excel
Contingency Table -CountIF - Excel
Published: 2016/05/22
Channel: Five Element Analytics
C 011 Calculating relative frequencies from a contingency table
C 011 Calculating relative frequencies from a contingency table
Published: 2013/01/11
Channel: John Wood
4 2 3 Two Way Tables : Contingency Tables
4 2 3 Two Way Tables : Contingency Tables
Published: 2012/08/07
Channel: R Backman
Working with Two-Way (Contingency) Tables in R
Working with Two-Way (Contingency) Tables in R
Published: 2016/09/02
Channel: Thomas Robacker
Use Chi-Square (X^2) Test for Contingency Table. TI 84. Stats 160 Final Review 34A
Use Chi-Square (X^2) Test for Contingency Table. TI 84. Stats 160 Final Review 34A
Published: 2012/12/14
Channel: mcstutoringstats
Contingency tables percentages
Contingency tables percentages
Published: 2013/08/05
Channel: Sea Otter
Contingency Tables with R
Contingency Tables with R
Published: 2016/06/12
Channel: Gilles Lamothe
Mosaic Plots and Contingency Tables
Mosaic Plots and Contingency Tables
Published: 2014/01/06
Channel: JMPSoftwareFromSAS
STATA Tutorials: Two- and Three-way Contingency Tables
STATA Tutorials: Two- and Three-way Contingency Tables
Published: 2012/07/02
Channel: Methodology LSE
Crosstabs and contingency tables (SPSS)
Crosstabs and contingency tables (SPSS)
Published: 2015/03/06
Channel: Oxford Academic (Oxford University Press)
Contingency Table in Excel a.k.a. pivot table
Contingency Table in Excel a.k.a. pivot table
Published: 2015/05/03
Channel: Stephanie Glen
Calculating Expected Counts in Contingency Tables by Marginal Proportions and Totals
Calculating Expected Counts in Contingency Tables by Marginal Proportions and Totals
Published: 2014/07/08
Channel: Eric Cai
Chi Square: Contingency Tables
Chi Square: Contingency Tables
Published: 2012/11/22
Channel: onlinestatbook
Categorical Variables & Contingency Tables in R
Categorical Variables & Contingency Tables in R
Published: 2017/03/22
Channel: Nathan Cole
Crosstabulation or Contingency Table Creation in StatCrunch
Crosstabulation or Contingency Table Creation in StatCrunch
Published: 2015/06/04
Channel: mpmyersphd
Calculating Chi-square for Contingency Tables (Crosstabulation)
Calculating Chi-square for Contingency Tables (Crosstabulation)
Published: 2013/08/08
Channel: Alan Neustadtl
Calculating Odds & Odds Ratios for Contingency Tables (Crosstabulation)
Calculating Odds & Odds Ratios for Contingency Tables (Crosstabulation)
Published: 2013/08/08
Channel: Alan Neustadtl
Contingency table.mp4
Contingency table.mp4
Published: 2013/05/11
Channel: Richard Cook
Find Critical Value of Chi-square for Two-way Contingency Table using StatCrunch
Find Critical Value of Chi-square for Two-way Contingency Table using StatCrunch
Published: 2017/02/25
Channel: The Stats Files - Dawn Wright Ph.D.
2 by 2 Contingency Table Analysis (Pearson Chi-Square) - SPSS (part 2)
2 by 2 Contingency Table Analysis (Pearson Chi-Square) - SPSS (part 2)
Published: 2011/09/20
Channel: how2stats
Mth120 Section 4.4 - Contingency Tables and Association
Mth120 Section 4.4 - Contingency Tables and Association
Published: 2015/09/11
Channel: Dan Kernler
Contingency Table in Excel
Contingency Table in Excel
Published: 2016/10/26
Channel: MathIsGreatFun
StatCrunch - Chi Square Contingency Table Example
StatCrunch - Chi Square Contingency Table Example
Published: 2014/04/03
Channel: DrCraigMcBridePhD
Contingency table  Completing
Contingency table Completing
Published: 2016/08/24
Channel: Joel Speranza Math
Statdisk Section 11-2: Using Statdisk for Contingency Tables
Statdisk Section 11-2: Using Statdisk for Contingency Tables
Published: 2017/01/09
Channel: Triola Stats
part 6 more contingency table and degrees of freedom
part 6 more contingency table and degrees of freedom
Published: 2016/12/03
Channel: Alex Holsinger
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WIKIPEDIA ARTICLE

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In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business intelligence, engineering and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The term contingency table was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation",[1] part of the Drapers' Company Research Memoirs Biometric Series I published in 1904.

A crucial problem of multivariate statistics is finding (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of the conditional independences are revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can use information theory concepts, which gain the information only from the distribution of probability, which can be expressed easily from the contingency table by the relative frequencies.

Example[edit]

Suppose that we have two variables, sex (male or female) and handedness (right or left handed). Further suppose that 100 individuals are randomly sampled from a very large population as part of a study of sex differences in handedness. A contingency table can be created to display the numbers of individuals who are male and right handed, male and left handed, female and right handed, and female and left handed. Such a contingency table is shown below.

Handed-
ness
Gender
Right handed Left handed Total
Male 43 9 52
Female 44 4 48
Total 87 13 100

The numbers of the males, females, and right- and left-handed individuals are called marginal totals. The grand total (i.e. the total number of individuals represented in the contingency table) is the number in the bottom right corner.

The table allows us to see at a glance that the proportion of men who are right handed is about the same as the proportion of women who are right handed although the proportions are not identical. The significance of the difference between the two proportions can be assessed with a variety of statistical tests including Pearson's chi-squared test, the G-test, Fisher's exact test, and Barnard's test, provided the entries in the table represent individuals randomly sampled from the population about which we want to draw a conclusion. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), we say that there is a contingency between the two variables. In other words, the two variables are not independent. If there is no contingency, we say that the two variables are independent.

The example above is the simplest kind of contingency table, a table in which each variable has only two levels; this is called a 2 × 2 contingency table. In principle, any number of rows and columns may be used. There may also be more than two variables, but higher order contingency tables are difficult to represent visually. The relation between ordinal variables, or between ordinal and categorical variables, may also be represented in contingency tables, although such a practice is rare.

Standard contents of a contingency table[edit]

  • Multiple columns (historically, they were designed to use up all the white space of a printed page). Where each row refers to a specific sub-group in the population (e.g., men), the columns are sometimes referred to as banner points or cuts (and the rows are sometimes referred to as stubs).
  • Significance tests. Typically, either column comparisons, which test for differences between columns and display these results using letters, or, cell comparisons, which use color or arrows to identify a cell in a table that stands out in some way (as in the example above).
  • Nets or netts which are sub-totals.
  • One or more of: percentages, row percentages, column percentages, indexes or averages.
  • Unweighted sample sizes (i.e., counts).

Measures of association[edit]

The degree of association between the two variables can be assessed by a number of coefficients. The simplest, applicable only to the case of 2×2 contingency tables, is the phi coefficient defined by

where χ2 is computed as in Pearson's chi-squared test, and N is the grand total of observations. φ varies from 0 (corresponding to no association between the variables) to 1 or −1 (complete association or complete inverse association), provided it is based on frequency data represented in 2 × 2 tables. Then its sign equals the sign of the product of the main diagonal elements of the table minus the product of the off–diagonal elements. φ takes on the minimum value −1.00 or the maximum value of 1.00 if and only if every marginal proportion is equal to .50 (and two diagonal cells are empty).[2]

Alternatives include the tetrachoric correlation coefficient (also only applicable to 2 × 2 tables), the contingency coefficient C, and Cramér's V.

C suffers from the disadvantage that it does not reach a maximum of 1 or the minimum of −1; the highest it can reach in a 2 × 2 table is 0.707; the maximum it can reach in a 4 × 4 table is 0.870. It can reach values closer to 1 in contingency tables with more categories. It should, therefore, not be used to compare associations among tables with different numbers of categories.[3] Moreover, it does not apply to asymmetrical tables (those where the numbers of row and columns are not equal).

The formulae for the C and V coefficients are:

and

k being the number of rows or the number of columns, whichever is less.

C can be adjusted so it reaches a maximum of 1 when there is complete association in a table of any number of rows and columns by dividing C by (recall that C only applies to tables in which the number of rows is equal to the number of columns and therefore equal to k).

The tetrachoric correlation coefficient assumes that the variable underlying each dichotomous measure is normally distributed.[4] The tetrachoric correlation coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories."[5] The tetrachoric correlation should not be confused with the Pearson product-moment correlation coefficient computed by assigning, say, values 0 and 1 to represent the two levels of each variable (which is mathematically equivalent to the phi coefficient). An extension of the tetrachoric correlation to tables involving variables with more than two levels is the polychoric correlation coefficient.

The lambda coefficient is a measure of the strength of association of the cross tabulations when the variables are measured at the nominal level. Values range from 0 (no association) to 1 (the theoretical maximum possible association). Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.

The uncertainty coefficient is another measure for variables at the nominal level.

The values range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.

See also[edit]

  • Confusion matrix
  • The pivot operation in spreadsheet software can be used to generate a contingency table from sampling data.
  • TPL Tables is a tool for generating and printing cross tabs.
  • The iterative proportional fitting procedure essentially manipulates contingency tables to match altered joint distributions or marginal sums.
  • The multivariate statistics in special multivariate discrete probability distributions. Some procedures used in this context can be used in dealing with contingency tables.

References[edit]

  1. ^ Karl Pearson, F.R.S. (1904). Mathematical contributions to the theory of evolution (PDF). Dulau and Co. 
  2. ^ Ferguson, G. A. (1966). Statistical analysis in psychology and education. New York: McGraw–Hill.
  3. ^ Smith, S. C., & Albaum, G. S. (2004) Fundamentals of marketing research. Sage: Thousand Oaks, CA. p. 631
  4. ^ Ferguson.
  5. ^ Ferguson, p. 244

Further reading[edit]

External links[edit]

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