How to Report Phi and Cramér's V in APA 7

You run the chi-square, it comes out significant, and the reviewer asks for an effect size. You open the output and find three coefficients stacked in the same table — phi, Cramér's V and the contingency coefficient — with three different values and not a word about which one goes into the manuscript. Here is which one your table shape calls for, how each symbol is written, and why the threshold that turns your number into small or medium moves with the degrees of freedom. The test that produces those coefficients is covered in how to report a chi-square in APA 7.

Phi for 2×2 tables, Cramér's V for everything else

The rule fits on one line: two rows and two columns, phi; more than two on either side, Cramér's V. Here it is with numbers. Two hundred and forty undergraduates screened for anxiety with the GAD-7 at a cut-off of 10: 56 of the 140 women (40.0%) and 24 of the 100 men (24.0%) scored above it. The test gives χ²(1, N = 240) = 6.72, p = .010, and the effect size is φ = the square root of 6.72/240 = .17. Because the table is 2×2, Cramér's V in that same output is also .17: these are not two results, they are one result with two names.

Once the table grows, the two part company. Four hundred and twenty patients from a mental health service, three admission diagnoses (depression, anxiety and adjustment disorder) and four discharge outcomes (recovered, improved, unchanged, deteriorated). The test gives χ²(6, N = 420) = 34.57, p < .001, and Cramér's V is the square root of 34.57/(420 × 2) = .20. That 2 in the denominator is df*, the smaller degrees of freedom of the table: min(3 − 1, 4 − 1) = 2. Phi is no use here: as soon as both sides of the table go past two categories it can exceed 1 — in a 3×4 it runs up to 1.41 — and stops reading like a correlation.

Both are interpreted the same way: 0 is independence and 1 is perfect association, and V can actually reach 1 whenever the table allows it. What they add is not significance — the χ² already settled that — but magnitude, which is why APA 7 asks for them. The gap between the two questions shows up best when you exaggerate it: with 5,000 cases, that same 3×4 table at V = .06 would give χ²(6, N = 5,000) = 36.00 and a p below .001. Significant, and for practical purposes nothing.

The same output carries a third coefficient, the contingency coefficient (C), and this is not a three-way tie. C can never reach 1 and its ceiling depends on the size of the table, so two values computed on differently shaped tables are not comparable with each other. Cramér's V was devised precisely to fix that (Cramér, 1946). Leave C where it is and do not report it.

V is italicized, phi is not, and neither takes a leading zero

The rule is not a stylistic whim, it is the same one that governs the whole Results section: symbols that are Latin letters are italicized, symbols that are Greek letters stay upright. Cramér's V is a Latin vee, so it is written V = .20, in italics. Phi is a Greek letter, so it is written φ = .17, upright, exactly like χ², η²_p, ω and α. Alongside them, in italics, live M, SD, r, t, F, N and d. The full list, with the decimals and zeros each index takes, is in APA 7 italics, decimals and leading zeros.

Neither of them takes a zero before the decimal point, for the usual reason: the leading zero goes only on quantities that can exceed 1. Phi and V are bounded between 0 and 1, so they are written .17 and .20, never 0.17 or 0.20. If you report phi with a sign — you will see below why it sometimes carries one — nothing changes: φ = −.17, no zero. Two decimals is standard and is what a reviewer expects; use three only when the value is so small that two would round it to .00.

The symbol breaks with surprising ease. Type the real φ, the Greek character (U+03C6), and not an italic f or an f in the Symbol font: the second looks fine on your screen and reaches the journal's PDF as a stray f or an empty box. SPSS labels the row Phi, spelled out, and that word is not the symbol. Capital Φ is not it either: in statistics it denotes something else, the normal distribution function. And V stands alone, with no subscript — V_c circulates, but APA 7 does not ask for it.

Here is the whole line with every piece in place: χ²(1, N = 240) = 6.72, p = .010, φ = .17. Notice two details that slip when the sentence is retyped by hand. Chi-square is the exception on degrees of freedom: it carries N inside the brackets too, which neither t nor F does. And that symbol is the Greek chi, χ, not a Latin ex; an X² in a Results section is the cheapest signal that the paragraph was typed without opening the manual.

V's thresholds depend on min(rows − 1, columns − 1)

The three numbers everybody quotes — .10 small, .30 medium, .50 large — are Cohen's (1988), but they do not belong to V: they belong to w, his index for contingency tables, computed as the square root of χ²/N. The relationship between the two is direct: V = w divided by the square root of df*. Since the statistic is divided by that root, the thresholds have to be divided by the same one. That is where the table worth keeping to hand comes from. At df* = 1: .10, .30 and .50. At df* = 2: .07, .21 and .35. At df* = 3: .06, .17 and .29. At df* = 4: .05, .15 and .25.

Go back to the 3×4 table of 420 patients. Here w is the square root of 34.57/420 = .29, one hundredth below Cohen's medium. V is .20 and, against the thresholds that belong to it — df* = 2, medium at .21 — it sits exactly the same distance away. These are not two rival readings: it is one reading with the square root of 2 divided out of both the statistic and the cut-offs, which is why they agree. What does change the conclusion is reading that .20 against .10, .30 and .50, which turns it into a small effect and throws away half the finding. When somebody writes that their V = .20 indicates a weak association without saying how many rows and columns the table had, they are not reporting, they are guessing.

The expensive mistake is confusing df* with the test's degrees of freedom, and it shows up as soon as a table is long on one side only. Three hundred students split between two teaching modes, face-to-face and online, classified by main presenting problem across five categories: χ²(4, N = 300) = 16.28, p = .003, V = .23. The test has 4 degrees of freedom, but df* = min(2 − 1, 5 − 1) = 1, so the thresholds that apply are .10, .30 and .50, and V = .23 is a small-to-medium effect. Read off the df* = 4 row (.05, .15 and .25), the same number would have been described as nearly large. Same datum, opposite adjectives.

These thresholds are conventions, not laws, and Cohen himself offered them as a last resort for when there is nothing better to compare against. Here there is: the percentages. In the table of 420 patients, discharge by recovery ran at 22.0% in depression, 34.4% in anxiety and 51.8% in adjustment disorder, and that gradient tells a clinician a good deal more than a .20 does. If your journal wants an interval, V has one: with these data, 95% CI [.12, .26], computed from the non-central chi-square (in R, effectsize::cramers_v(adjust = FALSE) returns it; left at its default that package applies the bias correction covered below and hands you a different number). An interval reaching .26 and dropping to .12 says more about your precision than any label.

Phi's ceiling, its sign, and when the odds ratio says more

Phi has a second definition that explains its oddities: it is the Pearson correlation between two dichotomous variables coded 0 and 1. With the GAD-7 table, (56 × 76 − 84 × 24) divided by the square root of (140 × 100 × 80 × 160) gives 2240/13386.56 = .17, the same value that came out of the square root of χ²/N. The correlation version carries a sign and the chi-square version does not, and that sign depends on the order in which you laid out the rows and columns. SPSS prints it for 2×2 tables and so does psych::phi(). If you report it, say which category you coded as 1; if not, drop the sign and describe the direction in words.

Phi's ceiling is not 1 unless the two variables split in the same place. With this study's margins — 58.3% women and 33.3% of cases above the cut-off — the most extreme table those totals allow would give φ = .60. That is the attainable maximum, not 1, so the observed .17 is 28% of the ceiling rather than 17% of the way to a perfect association. It is a structural limit that no amount of effect can get around, which is why the margins are worth reporting: without them, the reader has nothing to measure your coefficient against.

A small phi can sit happily beside a difference nobody would call small. Here φ = .17 describes 40.0% against 24.0%: sixteen percentage points apart. When the question is clinical or epidemiological, the odds ratio and the risk ratio carry that far better — OR = 2.11, 95% CI [1.19, 3.73]; RR = 1.67 — because they speak in the units decisions are made in. Reporting phi and the odds ratio together is not redundancy, they answer different questions; how each is computed and interpreted is in odds ratio and relative risk in clinical research.

One detail that throws tables out: phi is computed on the chi-square without the continuity correction. With these data, Yates's correction drops the statistic from 6.72 to 6.02 and p from .010 to .014, and if you carry that corrected value into the formula, phi falls from .17 to .16. Packages disagree about which they use by default, so state which version you are reporting. The same holds when the p comes from a Fisher exact test because of sparse cells: reporting the exact p with phi beside it is perfectly legitimate, but say so, because a reader trying to rebuild the coefficient from your p will not land on your number.

If you only have the chi-square, the N and the table size, you can convert chi-square to Cramér’s V exactly.

When neither of them applies

If both measurements come from the same people at two time points, phi and V describe nothing useful. They measure association between two variables, and what you have there is one variable measured twice: the test is McNemar and the effect size comes from the discordant cells, not from the whole table. Pasting a Cramér's V underneath a McNemar is one of the commonest errors and one of the easiest to spot; the correct report is in how to report McNemar's test in APA 7.

If both variables are ordered, V throws that information away. Education in three bands crossed with treatment adherence in three bands is not the same thing as eye color crossed with car brands, yet V treats them identically: shuffle the categories and it returns the same value. That calls for Kendall's tau-b, Goodman and Kruskal's gamma or a Spearman correlation, which also carry a sign and tell you whether the association rises or falls. V can never say that.

If you only have one variable, there is no association to measure. A goodness-of-fit chi-square — checking whether those 420 discharges fall out the way the literature said they would — compares observed frequencies against theoretical ones, and the index there is Cohen's w, the square root of χ²/N, with .10, .30 and .50 undivided. Phi and V need two crossed variables. If your table has a single row of data, those cells in the output are empty for a reason.

And V, where it does apply, does not say where the effect is. Adjusted standardized residuals do (Haberman, 1973). In the table of 420 patients, four cells clear the 2.87 that a Bonferroni correction for twelve cells sets: fewer recoveries than expected in depression (z = −4.02), more in adjustment disorder (z = 4.44), and in depression an excess of unchanged cases (z = 3.08) and of deteriorations (z = 2.88). Sparse cells also inflate V: Bergsma's (2013) bias correction takes this .20 down to .18, and with large tables and low frequencies the gap widens.

The paragraph, ready to copy

The full 2×2 fits in four sentences: “Above the GAD-7 cut-off were 40.0% of the women (56 of 140) and 24.0% of the men (24 of 100). The association between sex and clinically relevant anxiety symptoms was significant, χ²(1, N = 240) = 6.72, p = .010, φ = .17, computed on the uncorrected statistic, with OR = 2.11, 95% CI [1.19, 3.73]. Given these margins (58.3% women and 33.3% of cases above the cut-off), the highest value φ could reach is .60. All expected frequencies exceeded 33.” Nobody has to ask you for anything else.

The larger table needs one more sentence, the residuals one: “Discharge outcome was distributed differently across admission diagnoses, χ²(6, N = 420) = 34.57, p < .001, Cramér's V = .20, 95% CI [.12, .26]. Discharge by recovery ran at 22.0% in depression (33 of 150), 34.4% in anxiety (55 of 160) and 51.8% in adjustment disorder (57 of 110). Adjusted standardized residuals located the effect in four cells, using |z| > 2.87 (Bonferroni for 12 cells): depression with fewer recoveries (z = −4.02), more unchanged cases (z = 3.08) and more deteriorations (z = 2.88), and adjustment disorder with more recoveries (z = 4.44). No expected frequency fell below 7.8. With df* = 2, the reference values are .07, .21 and .35, so the association is medium in magnitude.”

The table carries the frequencies with the row percentage in brackets, the adjusted residuals flagged, and a note saying three things: what df* is, which coefficient V is, and the criterion the residuals were flagged with. For instance: “Note. Percentages are row percentages. V = Cramér's V; df* = min(rows − 1, columns − 1) = 2. Adjusted standardized residuals with |z| > 2.87 (Bonferroni criterion for 12 cells) are marked with an asterisk.” If you have the χ² and the N but would rather not reopen the output for the coefficient, paste the whole line into the effect size from text extractor: from a 2×2 table it returns φ, and with df > 1 it returns Cohen's w, which you divide by the square root of df* to reach V.

Four things worth deleting the moment they appear. A significant chi-square with no coefficient at all, which is what prompts the reviewer's request in the first place. A 0.20 with its leading zero, or an upright V beside an italic φ, which is the same rule inverted. A phi reported on a 3×4 table, where it can exceed 1. And a V called small or large without a word about the shape of the table, which is the quiet way of applying somebody else's threshold. If you are staring at the output and cannot tell which row is which, start by counting rows and columns: the shape of the table picks the coefficient, and the coefficient picks the threshold.

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Frequently asked questions

How do you report Cramér's V in APA 7?

Straight after the chi-square, in the same sentence, italicized and with no leading zero: χ²(6, N = 420) = 34.57, p < .001, V = .20. Two decimals are enough. Say what shape the table is, or state df* = min(rows − 1, columns − 1) outright, because the interpretive thresholds depend on that number. If the journal wants a confidence interval, V takes one: 95% CI [.12, .26] in this example.

When do you use phi and when Cramér's V?

Phi when the table is 2×2, Cramér's V when either side has three categories or more. In a 2×2 table the two return the same number, so it makes no difference which you report as long as you name it correctly. Once both sides go past two categories phi can exceed 1 and stops reading like a correlation, so only V works there. The contingency coefficient that appears in the same output is not reported: its maximum depends on the size of the table.

Is Cramér's V italicized?

Yes, because it is a Latin letter: V = .20. Phi is not, because it is a Greek letter and stays upright, exactly like χ², η²_p, ω and α: φ = .17. It is the same rule that italicizes M, SD, r, t and F. Type the genuine Greek character φ as well, not an italic f, which breaks on conversion to PDF.

What counts as a large Cramér's V?

It depends on the shape of the table, and this is the step almost everybody skips. The thresholds .10, .30 and .50 belong to Cohen's (1988) w and apply to V only when df* = min(rows − 1, columns − 1) equals 1. For anything else you divide them by the square root of df*: at df* = 2 they become .07, .21 and .35; at df* = 3, .06, .17 and .29; at df* = 4, .05, .15 and .25. They are conventions rather than laws, and describing the percentages helps more than the label does.

Do phi and Cramér's V take a leading zero?

No. The leading zero goes only on indices that can exceed 1, and both of these are bounded between 0 and 1. They are written φ = .17 and V = .20, never 0.17 or 0.20. It is the same rule that applies to p, r, α, ω and η²_p, and the one that does not apply to M, SD or B, which do take it when the value calls for it.

Can phi be negative?

Yes, if you compute it as a correlation between two dichotomous variables coded 0 and 1, which is what SPSS and psych::phi() do for 2×2 tables. Computed as the square root of χ²/N it always comes out positive. The sign depends on the order in which you laid out the rows and columns, so it is arbitrary unless the categories have a natural direction: either say which category you coded as 1, or drop the sign and describe the direction in words. Cramér's V is never negative.

With the coefficient your table shape calls for, V italicized, φ upright, neither carrying a leading zero and the threshold divided by the square root of df*, your effect size is written the way APA 7 asks. When the doubt comes earlier — whether that contingency table really answers your question, or whether the design called for something else — it is cheaper to settle before analyzing than after: in statistical consulting we go through the full analysis plan and put in writing what gets reported in each table.