Linguistics 483: Notes

A. C. Brett acbrett@uvic.ca Department of Linguistics University of Victoria Clearihue C139

Example

Null Hypothesis

The process of establishing whether or not the crosstabulated variables are related normally consists of first adopting the hypothesis that there is no relationship among them; that is, it is first assumed that the variables are independent or they are not contingent upon each other. This assumption is called the null hypothesis and is denoted by H₀.

A reasonable null hypothesis to adopt in assessing whether or not the Affix Onset Pitch Change and Affix variables in the foregoing table are related might be as follows:

If the data measured on the variables provide sufficiently compelling evidence that the null hypothesis is not true, then H₀ can be rejected. The alternative hypothesis, that the variables are in fact dependent or contingent upon one another, may then be claimed. This alternative to H₀ is usually denoted by A.

A reasonable alternative to the hypothesis of the independence of the Affix Onset Pitch Change and Affix variables in the foregoing example might be as follows:

Note that, if the evidence provided by the data on the variables is not sufficiently strong that the null hypothesis can be rejected, one cannot claim that H₀ is necessarily true. H₀ might in fact be false; it is only the case that the evidence available is not sufficiently compelling to enable one to reject it.

Test Statistic

The decision to reject H₀ and claim A, or to not reject H₀ if the evidence is not sufficiently compelling, is customarily based on the value of a test statistic such as c² (usually read as "chi square" with "ch" pronounced as /k/). A test statistic normally measures the extent to which the evidence provided by the data differs from what one would expect if the null hypothesis were true. If the discrepancy indicated by the value of the test statistic is sufficiently large, one normally rejects H₀ and claims A.

The conclusion that the value of a test statistic is large enough that one can reject H₀ is based upon the behaviour of the statistic as the data differ from what one would expect if H₀ were true. If certain conditions are satisfied, which include the condition that H₀ is true, a test statistic such as c² has a known behaviour.

The behaviour of a test statistic is typically described by a probability distribution function. A probability can therefore be associated with a given value of the statistic. This probability measures the likelihood of obtaining by chance alone a value of the test statistic as large as, or larger than the value observed if the null hypothesis is true.

Significance Level

The decision to either reject or not reject H₀ is customarily based upon the probability associated with the value obtained for the test statistic. One normally selects a probability, called the significance level and denoted a, corresponding to the largest chance of error one is willing to tolerate in rejecting a true null hypothesis.

Rejection of a true null hypothesis is referred to as a Type I error. The complementary error, namely, the failure to reject a null hypothesis that is in fact false, is described as a Type II error.

Customarily, a significance level of a = 0.05 or a = 0.01 is selected. Adopting a = 0.05 means that one is willing to run a 5%, or a one in twenty chance of committing a Type I error, that is, the chance of being wrong in rejecting a true null hypothesis.

If the probability associated with the test statistic value is less than or equal to the selected a, then one will normally reject H₀, and claim A, because the chance of committing a Type I error is less than the greatest chance of error one is willing to tolerate.

c² Formula

• r and c denote the number of rows and columns, respectively, in the table;
• S_i^r represents summation over rows from i = 1 to i = r, and S_j^c represents summation over columns from j = 1 to j = c;
• o_ij is the count (frequency) observed for the cell in the i^th row and j^th column of the table; and,
• e_ij is the count (frequency) expected for the corresponding cell under the hypothesis that the crosstabulated variables are independent.

The c² statistic is thus a measure of the extent to which the observed cell counts depart from the cell counts expected under H₀. The square root of c² is something like a distance between observed and expected cell counts.

Row and Column Values

For example, the column variable Y in the foregoing table is the affix of the numbers uttered by the participants in the experiment. This variable has two values, namely, those labelled "-teen" and "-ty". Thus, the number of columns c in the table is 2, with the row index i = 1 corresponding to the value label "-teen", and the index i = 2 corresponding to the value label "-ty".

The row variable X in the table is the affix onset pitch change, the values on which have been computed from the fundamental frequencies measured at the onset of the utterance of the affix, and at the midpoint of the utterance. If the frequency at the midpoint is less than or equal to the initial frequency, the value "falling" is assigned to the variable. If the frequency at the midpoint is higher than the initial frequency, the value "rising" is assigned. The number of rows r in the table is therefore 2. The column index j = 1 corresponds to the value label "falling", while the index j = 2 corresponds to the value label "rising".

Cell Counts

For example, the cell count in the foregoing table are as follows:

The cell count for the first row and first column of the foregoing table, o₁₁ = 69, is the number of "-teen" affixes uttered with a falling onset pitch, and in the second row, o₂₁ = 7 is the number of "-teen" affixes uttered with a rising onset pitch. In the second column of the table, o₁₂ = 53 and o₂₂ = 15 are the number of "-ty" affixes uttered with falling and rising onset pitch, respectively.

Note that some variability in cell counts is to be anticipated. This variability can be attributed to sampling effects, to measurement uncertainties, and to the variability inherent in human behaviour. Hence, if the experiment were repeated, different cell counts likely would be obtained.

In fact, it is assumed that the cell counts have a normal distribution; that is, one might imagine that if the experiment were repeated a number of times, the counts for each of the cells would be normally distributed. The test statistic actually has a c² distribution only if the cell counts are normally distributed.

Expected Counts

• T_i. is the total count for the i^th row of the table;
• T._j is the total count for the j^th column; and,
• T.. is the total count for the table.

Calculation of the expected count for each cell is illustrated in the foregoing table beside the title "Calculation." The resulting expected count is shown beside the title "Expected Count." The calculations and results for each cell are as follows:

Thus, the expected count e₁₁ = 64.4 for the cell in the first row and first column of the table indicates that approximately 64 observations of falling pitch would be expected for the "-teen" affix if the onset pitch change were indeed independent of the affix. Similarly, the expected count e₂₁ = 11.6 for the cell in the second row and first column indicates that approximately 12 observations of rising pitch are to be expected for the "-teen" affix if the onset pitch change is independent of the affix.

Marginal Frequencies

The marginal distributions are just the frequency distributions that would be obtained for the row and column variables separately. The ratio T._j /T.. of the column marginal frequency T._j to the total T.. is the relative frequency f_j^Y of the j^th value on the variable Y; that is,

If the variables X and Y are independent, that is, the values of X do not depend on the values of Y, then the distribution of cell counts in each row of the table are determined by the relative frequencies f_j^Y for j = 1 to j = c. Thus, for each i for i = 1 to i = r, the total observations T_i. for the i^th row will be distributed among the c columns according to the column marginal distribution f_j^Y. The count or frequency expected for the cell in the j^th column and i^th row under the hypothesis of independence is then given by

For example, for the first row in the foregoing table, the total or marginal count T₁. = 122. Since the column marginal relative frequency values are f₁^Y = 76/144 = .528 and f₂^Y = 68/144 = .472, the counts for the first row of the table expected under the hypothesis of independence are e₁₁ = 122×.528 = 64.4 and e₁₂ = 122×.472 = 57.6 . The counts expected for the second row, assuming the independence of the two variables, are e₁₁ = 22×.528 = 11.6 and e₁₂ = 22×.472 = 10.4 because the total or marginal count T₂. = 22 .

Degrees of Freedom

The number of degrees of freedom, n, in an r × c contingency table can be calculated as follows:

This formula can be explained by considering that the row and column marginal counts are fixed. The cell counts, on the other hand, are free to vary. If the null hypothesis is true, the distribution of observations among cells is "random" and the variability can be attributed to such factors as sampling effects, measurement uncertainty, or variability inherent in the quantities being measured.

Cells counts, however, are constrained by the fixed marginal counts. Thus, cell counts for the i^th row are constrained to add up to the row total T_i. so that only c − 1 of the c cells in the i^th row are actually free to vary.Similarly, cell counts for the j^th column are constrained to add up to the column total T._j so that only r − 1 of the r cells in the i^th column are actually free to vary.

The number of cells in each row and column for which the counts are freely variable is therefore reduced by 1, which results in the formula cited above for calculating number of degrees of freedom, n, in an r × c contingency table. For the table used as an example here, r = 2 and c = 2. Hence, n = (2 − 1) × (2 − 1) = 1 × 1 = 1.

c² Value

Given the expected cell counts determined as described above, the c² value for the table in the example may be calculated as follows:

c² = (69 − 64.4)²/64.4 + (53 − 57.6)²/57.6 + (7 − 11.6)²/11.6 + (15 − 10.4)²/10.4 = 4.577 .

If the observed cell counts are independently normally distributed, and the null hypothesis is true, then the calculated value of 4.577 is from a c² distribution.

Since the table has one degree of freedom, he probability associated with the value c² = 4.577 is 0.032, which means that P₁(c² ³ 4.577) = 0.032. In other words, the probability of obtaining a value of c² greater than or equal to 4.577 by chance alone, if the null hypothesis is true, is 0.032 .

If a significance level a = 0.05 has been selected, the null hypothesis that the variables are independent can be rejected can be rejected because a ³ 0.032. The alternative that onset pitch change is dependent on affix can then be claimed.

Continuity Correction

The c² distribution is continuous. The distribution calculated on the basis of the finite number of different possible cell or interval counts in a contingency table, however, is discrete. The distribution of the c² value calculated from the cell counts of a table with one degree of freedom can differ substantially from the continuous c² distribution.

It is generally believed that this difference can lead to the inappropriate rejection of null hypotheses that are actually true, particularly when the expected counts for some cells are less than 5 or 10. To minimise the likelihood that true null hypotheses will be rejected, the magnitude of the difference between observed and expected cell counts is reduced by 0.5 . This adjustment is described as a "continuity correction."

Incorporation of the continuity correction yields the following formula for calculating c²:

where |o_ij − e_ij| represents the magnitude or absolute value of the difference between o_ij and e_ij.

Application of the continuity correction to the calculation of c² for the contingency table used as an example in these notes yields c² = 3.638 with an associated probability of 0.056 . Since this probability is greater than the significance level a = 0.05, one likely would not reject the null hypothesis that the variables involved are independent.

Fischer's Exact Test

An exact probability associated with the observed cell counts in a 2 × 2 table can be calculated on the basis of the number of different ways in which the cells might be populated given the fixed marginal counts. The following formula yields this exact probability:

where the exclamation point denotes the factorial function which is defined for any integer n such that

For example 5! = 5×4×3×2×1 = 120 . The probability of a table with the given marginal and observed cell counts is thus the ratio of the product of the factorials of the marginal counts to the product of the factorials of the total count and of the observed cell counts.

For the 2 × 2 table used here as an example, the exact probability formula yields a value of p = 0.038 . Since p < a = 0.05, one likely would reject the null hypothesis and claim the alternative that the variables are dependent.

It would thus appear that, in this example, the continuity correction described above yields a result which is too conservative. The unadjusted c² consequently seems to produce the more appropriate result, and rejection of the null hypothesis at the 5% level is a reasonable decision in this case.