Statistics 2nd Year Chapter 15 Association Notes (ICS Part-II)

Chapters 12-14 dealt with quantitative (numerically measurable) variables — means, proportions, and the strength of linear relationships between two measured variables. Many real observations, however, are only qualitative: people classified as satisfied/neutral/dissatisfied with their jobs, manufactured items graded excellent/good/poor/scrap, households owning no car/one car/two-or-more cars. This chapter develops the parallel machinery for such categorical data, called attributes. Just as correlation measures the strength of a linear relationship between two quantitative variables, association measures the strength of relationship between two qualitative attributes.

The chapter builds from the simplest case — two attributes, each either present or absent, forming a 2×2 table — through Yule's coefficient of association Q, up to the general r x c contingency table and the chi-square test of statistical independence, the workhorse test for categorical data used throughout applied statistics. It closes with Spearman's rank correlation coefficient, which measures the degree of agreement between two rankings of the same set of objects — a natural bridge back to the correlation ideas of Chapter 14, but applied to ranks rather than raw measurements.

Learning Objectives

• Distinguish an attribute (qualitative variable) from a quantitative variable
• Define class, class frequency, dichotomy, positive/negative attributes, and ultimate class frequency
• Test whether the data on two attributes are consistent
• State the condition for independence of two attributes and compute expected frequencies under independence
• Define positive association, negative association, and complete association/disassociation
• Compute and interpret Yule's coefficient of association, Q
• Construct and interpret an r x c contingency table
• State and apply the chi-square test of statistical independence, including Yates' correction for 2×2 tables
• Compute Pearson's coefficient of mean square contingency, C, and its maximum possible value
• Compute and interpret Spearman's coefficient of rank correlation

Key Concepts

15.1 Multinomial Populations

When each element of a population is assigned to one, and only one, of more than two attribute categories, the population is called a multinomial population. This generalises the binomial idea (success/failure, two categories) to any number of mutually exclusive categories — for example, classifying manufactured items as excellent, good, poor, or scrap.

15.2 Attribute (Qualitative Variable)

An attribute is a characteristic that varies only in QUALITY, not quantity, from one individual to another — marital status, education level, blindness, smoking, beauty. An attribute cannot be measured numerically; data on it is obtained simply by noting whether each object possesses the attribute, and counting how many do and do not. A class is a set of objects sharing a given characteristic, and a class frequency is the number of observations falling into a class. Dichotomy is the process of dividing objects into two mutually exclusive, complementary classes according to whether or not they possess a particular attribute; classification can continue indefinitely by further dichotomising on additional attributes.

Notation: capital letters A, B, … denote presence of attributes; Greek letters alpha, beta, … denote their absence ('not A', 'not B'). Class frequencies are written in brackets: (A) = number possessing A; (alphaB) = number possessing B but not A. Classes represented purely by positive attributes (A, B, AB) are positive classes; purely negative attributes (alpha, beta, alphabeta) are negative classes; mixed classes (Abeta, alphaB) are contrary classes. The order of a class equals the number of attributes specifying it (order 1: (A), (alpha); order 2: (AB), (Abeta), (alphaB), (alphabeta)); the class of order zero is the total n. For k attributes, the total number of class frequencies (including n) is 3^k, and the ultimate class frequencies — those of the HIGHEST order — number 2^k.

Every class frequency can be expressed as the sum of its two subclasses from further dichotomy, e.g. n=(A)+(alpha), (A)=(AB)+(Abeta). Data are consistent if no class frequency computed this way is negative; the necessary and sufficient condition for consistency is that no ULTIMATE class frequency is negative. Inconsistency may arise from wrong counting, arithmetic errors, or misprints — but consistency itself is no guarantee that the counting was accurate.

15.3 Independence of Attributes

Two attributes A and B are independent if the proportion possessing both equals the product of the proportions possessing each separately: (AB)/n = [(A)/n] x [(B)/n], equivalently (AB) = (A)(B)/n. This is the categorical-data analogue of statistical independence of events, P(A and B)=P(A)P(B). Under independence, every cell of the 2×2 table can be filled in purely from the row and column totals, using this same multiplication rule for each of the four cells.

15.4 Association of Attributes (Correlation of Qualitative Variables)

Two attributes are associated if they are NOT independent, i.e. (AB) != (A)(B)/n. Association is positive (the attributes are 'simply associated') if (AB) > (A)(B)/n — the attributes occur together more often than chance/independence would predict. Association is negative (the attributes are 'disassociated') if (AB) < (A)(B)/n — they occur together less often than chance would predict. Disassociation does NOT imply independence; it is a distinct, opposite-direction relationship.

Complete (perfect positive) association exists if neither attribute can occur without the other except possibly in one direction: if (A)=(B), all A's are B's and vice versa; if (A)<(B), all A's are B's; if (B)<(A), all B's are A's. Complete disassociation (perfect negative association) exists if none of the A's are B's and none of the alpha's are beta's — the two attributes are mutually exclusive.

Yule's coefficient of association, Q, measures the strength of association: Q = [(AB)(alphabeta) – (Abeta)(alphaB)] / [(AB)(alphabeta) + (Abeta)(alphaB)]. Q always lies between -1 and +1: Q=0 means independence, Q=+1 means complete association, Q=-1 means complete disassociation. Q is the categorical-data counterpart of the correlation coefficient r from Chapter 14.

15.5 Two Dimensional Count Data: Contingency Table

When attribute A has r categories and attribute B has c categories, a simple random sample produces an r x c contingency table (a term coined by Karl Pearson) of observed cell frequencies o_ij, i=1,…,r and j=1,…,c. The cell frequency is the count falling in a particular (row, column) combination. The i-th row total is o_i. = sum over j of o_ij; the j-th column total is o_.j = sum over i of o_ij; and the grand total is n = sum of all o_ij = sum of all row totals = sum of all column totals.

15.6 Test for Statistical Independence

The chi-square test of independence tests H0: the row attribute and column attribute are independent (P(Ai and Bj)=P(Ai)P(Bj) for all i,j) against H1: they are not independent for at least one cell. Under H0, the expected frequency for cell (i,j) is e_ij = (o_i. x o_.j)/n — the row total times the column total, divided by the grand total, exactly mirroring the independence condition of section 15.3. The test statistic is chi-square = sum over all cells of [(o_ij – e_ij)^2 / e_ij], which follows an approximate chi-square distribution with v=(r-1)(c-1) degrees of freedom for large n. Large discrepancies between observed and expected frequencies inflate chi-square; if observed and expected agree exactly, chi-square=0. The critical region for level of significance alpha is chi-square > chi-square_{v; 1-alpha} (upper tail only — large chi-square, not small, indicates departure from independence).

A commonly used rule of thumb requires each expected frequency e_ij to be at least 5 for the chi-square approximation to be adequate; if some cells have smaller expected frequencies, neighbouring rows or columns can be combined (pooled), which reduces the degrees of freedom by one for each such combination performed.

Yates' Correction for Continuity applies specifically to 2×2 tables (v=1 degree of freedom), improving the approximation of the discrete data to the continuous chi-square distribution: Adjusted chi-square = sum of [(|o-e|-0.5)^2 / e]. The correction reduces the value of chi-square (making rejection of H0 less likely), and should only be omitted when |o-e| is already less than 0.5.

Pearson's Coefficient of Mean Square Contingency, C = sqrt[chi-square / (n + chi-square)], measures the STRENGTH of an association once independence has been rejected, on a scale that (unlike raw chi-square) does not depend purely on sample size. Its maximum possible value depends on the table's dimensions: max C = sqrt[(q-1)/q], where q is the smaller of the number of rows or columns — so C should always be interpreted relative to this ceiling, not against a fixed universal scale like the correlation coefficient r.

15.7 Rank Correlation

Rank correlation measures the correlation between the RANKS (not the raw measurements) assigned to individuals on two variables — for instance, two judges each ranking the same n contestants. Spearman's coefficient of rank correlation, r_r, is the ordinary Pearson correlation coefficient computed on the two sets of ranks, and simplifies to r_r = 1 – [6.sum(d_i^2)] / [n(n^2-1)], where d_i = x_i – y_i is the difference between an individual's two ranks. Like r, r_r always lies between -1 and +1: r_r near +1 indicates strong agreement between the two rankings, r_r near -1 indicates strong disagreement (reversed order), and r_r near 0 indicates little or no relationship between the rankings. Spearman's method is especially useful when the underlying data are not available as precise numerical measurements but can still be meaningfully ranked, or when the assumptions needed for Pearson's r (e.g. approximate linearity, interval-scale data) are in doubt.

Important Definitions

Key Facts and Relations

Topic	Key Fact / Relation
Total / ultimate class frequencies (k attributes)	Total class frequencies = 3^k; Ultimate class frequencies = 2^k
Independence condition	(AB) = (A)(B)/n
Positive / negative association	Positive: (AB) > (A)(B)/n; Negative: (AB) < (A)(B)/n
Yule's coefficient of association, Q	Q = [(AB)(alphabeta) – (Abeta)(alphaB)] / [(AB)(alphabeta) + (Abeta)(alphaB)]
Expected frequency under independence (contingency table)	e_ij = (o_i. x o_.j) / n
Chi-square test statistic	chi-sq = sum_i sum_j [(o_ij – e_ij)^2 / e_ij], v = (r-1)(c-1)
Yates' continuity-corrected chi-square (2×2 tables)	Adjusted chi-sq = sum [(|o-e| – 0.5)^2 / e]
Pearson's coefficient of mean square contingency	C = sqrt[chi-sq / (n + chi-sq)]
Maximum possible value of C	max C = sqrt[(q-1)/q], q = smaller of (rows, columns)
Mean square contingency	phi^2 = chi-sq / n
Spearman's rank correlation coefficient	r_r = 1 – [6.sum(d_i^2)] / [n(n^2-1)], d_i = x_i – y_i (difference of ranks)
Critical region for independence test	Reject H0 if chi-sq > chi-sq_{v; 1-alpha} (upper tail only)

Diagrams

Association Patterns: Positive, Negative, Independent: A 3-panel grouped bar chart comparing the observed frequency (AB) against the expected-under-independence frequency (A)(B)/n for three worked cases: positive association ((AB) noticeably taller than expected), negative association ((AB) noticeably shorter than expected), and independence ((AB) equal to expected), illustrating how the sign of (AB)-(A)(B)/n determines the direction of association

Chi-Square Test for Independence (Example 15.6): A chi-square distribution curve with v=6 degrees of freedom, with the upper-tail rejection region beyond the critical value chi-sq=12.59 (at alpha=0.05) shaded, and the observed test statistic chi-sq=54.06 from the degree/hobby contingency table plotted well inside the rejection region, visually confirming the decision to reject the null hypothesis of independence

Spearman Rank Correlation (Example 15.9): A scatter plot of the six paired ranks (x_i, y_i) from Example 15.9 with a reference line y=x (perfect agreement) overlaid, showing how closely the two judges'/measurements' rankings track each other, illustrating the moderate positive rank correlation r_r=0.4857 obtained for this data

Short Questions & Answers

Long Questions & Answers

Explain the full logic of the chi-square test for statistical independence in a contingency table — the null and alternative hypotheses, the expected frequencies, the test statistic, and the role of degrees of freedom — illustrating each step using Example 15.6 (degree vs hobby).

What question does the chi-square test of independence answer, and what are H0 and H1?

It asks whether two categorical attributes, cross-classified in a contingency table (r rows by c columns of observed frequencies o_ij, a format due to Karl Pearson), are statistically related. H0 states the two attributes are independent — for every cell, the probability of falling into it equals the product of its row and column marginal probabilities, mirroring the multiplication rule for independent events. H1 simply states that independence fails for at least one cell, i.e. some association exists.

How is the expected frequency for each cell calculated under H0?

Observed frequencies cannot be compared to a fixed benchmark directly, so we first compute what each cell would look like if H0 were true, given the row and column totals actually observed. The expected frequency for cell (i,j) is e_ij = (row i total * column j total) / n — the independence condition (AB)=(A)(B)/n generalized from a simple 2×2 table to a full r x c table.

How is the chi-square test statistic calculated, and why does a large value count as evidence against H0?

Chi-square is the sum, over every cell, of (observed – expected)^2 / expected. Squaring prevents positive and negative deviations from cancelling out, and dividing by e_ij weights each discrepancy relative to how large a frequency was expected in that cell. If H0 were exactly true, chi-square would equal zero; the further observed frequencies drift from expected ones, the larger chi-square becomes, so only unusually large values, in the upper tail, count as evidence against H0.

What are the degrees of freedom for this test, and how does Example 15.6 apply the full procedure?

Degrees of freedom are v=(r-1)(c-1), since fixing the row and column totals leaves only (r-1)(c-1) interior cells free to vary. In Example 15.6, 492 candidates cross-classified by degree (3 categories) and hobby (4 categories) give v=(4-1)(3-1)=6; at alpha=0.05 the critical value is chi-square(6,0.95)=12.59. The computed chi-square works out to 54.06, far beyond 12.59, so H0 is rejected — degree and hobby are not independent.

Once independence is rejected, how is the strength of the association measured?

Pearson's coefficient of mean square contingency, C = sqrt[chi-square/(n+chi-square)], gives C = sqrt[54.06/(492+54.06)] = 0.315 for Example 15.6. This is interpreted against its maximum possible value for a table of that shape, sqrt[(q-1)/q], where q is the smaller table dimension; here q=3 (the degree categories), giving a maximum possible value of sqrt(2/3)=0.8165.

Compare Yule's coefficient of association (Q) for a simple 2×2 table with Spearman's coefficient of rank correlation (r_r), explaining what each measures, how each is computed, and when each would be the appropriate tool to use.

What kind of data is Yule's coefficient of association Q designed for, and how is it computed?

Q applies to the simplest categorical case: two attributes, each simply present or absent, forming a 2×2 table with frequencies (AB), (Abeta), (alphaB), (alphabeta) — both present, only the first, only the second, and neither. Q = [(AB)(alphabeta) – (Abeta)(alphaB)] / [(AB)(alphabeta) + (Abeta)(alphaB)], comparing the product of the 'agreement' cells against the product of the 'disagreement' cells.

How is Q interpreted, from -1 through 0 to +1?

Q=0 means the two attributes are exactly independent, i.e. (AB) equals what chance alone — the two marginal rates — would predict. Positive Q means the attributes occur together more often than chance would predict, reaching +1 only under complete association (possessing one guarantees the other). Negative Q means they occur together less often than chance, reaching -1 only under complete disassociation, where the attributes are essentially mutually exclusive.

What kind of data is Spearman's rank correlation r_r designed for, and how is it computed?

r_r is built for variables that can be ranked, such as two judges each ranking the same set of contestants. Each variable's n observations are converted into ranks from 1 to n, the difference d_i between an individual's two ranks is found, and r_r = 1 – [6*sum(d_i^2)] / [n*(n^2-1)]. Perfect agreement gives every d_i=0 and r_r=+1; a perfect reversal gives r_r=-1; no relationship gives r_r near zero.

When should Q be used instead of r_r, or vice versa?

Q is the appropriate tool when both variables are genuinely categorical, dichotomous attributes (yes/no, present/absent) rather than measurements that could be ranked — for example, vaccinated (yes/no) versus contracted a disease (yes/no). r_r is appropriate when the data consists of, or can be reduced to, an ordering of individuals along a scale — for example, comparing how closely two examiners' rankings of the same essays agree, even without any numerical scores being recorded.

Multiple Choice Questions (MCQs)

Quick Revision Summary

• Attribute = qualitative characteristic (present/absent) | Variable = quantitative, numerically measurable
• Dichotomy = splitting objects into two mutually exclusive classes on one attribute
• Order of class = number of attributes specifying it; ultimate class frequencies = highest order, count = 2^k
• Consistency: data are consistent iff no ULTIMATE class frequency is negative
• Independence: (AB) = (A)(B)/n | Positive association: (AB) > (A)(B)/n | Negative: (AB) < (A)(B)/n
• Yule's Q = [(AB)(ab)-(Ab)(aB)]/[(AB)(ab)+(Ab)(aB)]; Q=0 independent, Q=+1 complete association, Q=-1 complete disassociation
• Contingency table: r rows x c columns of observed o_ij; e_ij = (row total)(col total)/n
• Chi-square = sum[(o-e)^2/e], v=(r-1)(c-1); reject H0 (independence) if chi-square > critical value (upper tail only)
• Yates' correction (2×2 only): subtract 0.5 from |o-e| before squaring, to improve the continuous approximation
• Pearson's C = sqrt[chi-square/(n+chi-square)]; max C = sqrt[(q-1)/q], q = smaller of rows/columns
• Spearman's r_r = 1 – 6.sum(d_i^2)/[n(n^2-1)], d_i = rank difference; range -1 to +1, like ordinary r

Exam Tips

• Never confuse 'disassociation' with 'independence' — disassociation is a specific NEGATIVE relationship, not the absence of one
• Always check e_ij >= 5 in every cell before trusting the chi-square approximation; pool rows/columns if needed and reduce df accordingly
• Apply Yates' correction ONLY to 2×2 tables (v=1); it is not used for larger tables
• The chi-square rejection region is always in the UPPER tail only — there is no 'too good a fit' rejection region in this test
• When reporting C, always state it alongside its maximum possible value sqrt[(q-1)/q] for that table shape — a raw C value alone is not comparable across differently-shaped tables
• For rank correlation, first double-check that ranks (not raw scores) have been assigned consistently in the same direction (both high-to-low or both low-to-high) before computing d_i