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Here is a term paper on ‘Chi-Square Test’ especially written for school and college students.

The X^{2} (Greek letter X^{2} Pronounced as Ki-square) test is a method of evaluating whether or not frequencies which have been empirically observed differ significantly from those which would be expected under a certain set of theoretical assumptions.

**For example, suppose political preference and place of residence or nativity have been cross- classified and the data summarised in the following 2×3 contingency table. **

It is seen in the table that the proportions of urban people are 38/48 = 0.79, 20/46 = 0.34, and 12/18 = 0.67 (rounded to two decimals) for the three political parties in the country. We would then want to know whether or not these differences are statistically significant.

To this end, we can propose a null hypothesis which assumes that there are no differences among the three political parties in respect of nativity. This means that the proportions of urban and rural people should be expected to be the same for each of the three political parties.

On the basis of the assumption that the null hypothesis is correct, we can compute a set of frequencies that would be expected given these marginal totals. In other words, we can compute the number of persons showing preference for the Congress party whom we would expect on the basis of the above assumption to be urbanites and compare this figure with the one actually observed.

**If the null hypothesis is true, we may compute a common proportion as: **

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38 + 20 + 12/48 + 46 + 18 =70/112= 0.625

With this estimated proportion, we would expect 48 x (0.625) = 30 persons affiliated to Congress, 46 _{x} (0.625) = 28.75 persons affiliated to Janata Party and 18 x (0.625) = 11.25 persons affiliated to Lok Dal from out of the 70 urbanites. Subtracting these figures from the respective observed figures from the respective sizes of the three samples, we find 48 – 30 = 18 affiliated to Congress, 46 – 28.75 = 17.25 affiliated to Janata and 18 – 11.25 = 6.25 persons affiliated to Lok Dal from 42 persons from the rural areas.

These results are shown in the following table, where expected frequencies ar. shown in parenthesis.

To test the tenability of the null hypothesis we compare the expected and observed frequencies. The comparison is based on the following X^{2} statistic.

X^{2 }= Σ (O- E)^{2 }/E

where O stands for observed frequencies and E for the expected frequencies.

**Degrees of Freedom**:

The number of degrees of freedom means the number of independent constraints imposed on us in a contingency table.

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**The following example will illustrate the concept: **

Let us suppose that the two attributes A and B are independent in which case, the

expected frequency or the cell AB would be 40×30/60 = 20. Once this is identified, the frequencies of remaining three cells get automatically fixed. Thus, for cell, αB the expected frequency must be 40 – 20 = 20, similarly, for the cell AB it must be 30 – 20 = 10 and for αB it must be 10.

This means that for 2×2 table we have just one choice of our own, while we have no freedom in the remaining three cells. Thus, the degrees of freedom (df) can be calculated by the formula:

df – (c – 1) (r – 1)

where df stands for the degrees of freedom, c for the number of columns and r for the number of rows.

Thus, in 2 x 3 table (Table 18.54)

df = (3 – 1) (2 – 1) = 2 x 1 = 2

To use the chi-square test, the next step is to calculate the degrees of freedom: Suppose we have a 2 x 2 contingency table like the one in Fig. 1.

We know the row and the column totals r_{t}^{1} and r_{t }^{2}– and c_{t}^{1} and c_{t}^{2}. The number of degrees of freedom can be defined as the number of cell-values that we can freely specify.

In Fig. 1, once we specify the one value of Row 1 (denoted by check in the figure) the second value in that row and the values of second row (denoted by X) are already determined; we are not free to specify these because we know the row totals and column totals. This shows that in a 2 x 2 contingency table we are free to specify one value only.

**Procedure**:

**Computation for Chi-Square Test:**

**Level of Significance**:

As stated earlier, the chi-square test is used to examine whether the difference between observed and expected frequencies is due to the sampling fluctuations and as such insignificant or contrarily, whether the difference is due to some other reason and as such significant.

Before drawing the inference that the difference is significant researchers set up a hypothesis, often referred as a null hypothesis (symbolized as H_{o}) as contrasted with the research hypothesis (H_{1}) that is set up as an alternative to H_{o}.

Usually, although not always, the null hypothesis states that there is no difference between several groups or no relationship between variables, whereas a research hypothesis may predict either a positive or negative relationship.

In other words, null hypothesis assumes that there is absence of non-sampling errors and the difference is due to chance alone. Then the probability of the occurrence of such a difference is determined.

The probability indicates the extent of reliance that we can place on the inference drawn. The table values of chi-square are available at various probability levels. These levels are called levels of significance. We can find out from the table the values of chi-square at certain levels of significance.

Usually (in social sciences problem), the value of chi-square at 0.05 or .01 levels of significance from the given degrees of freedom is seen from the table and is compared with observed value of chi-square. If the observed value or y^{1} is more than the table value at 0.05, it means that the difference is significant.

**Chi-Square as a Test of Goodness of Fit****: **

In the previous section we used the chi-square as a test of independence; that is whether to accept or reject a null hypothesis. The x~ tests can also be used to decide whether there is a significant difference between an observed frequency distribution and a theoretical frequency distribution.

In this manner we can determine how good the fit is of the observed and expected frequencies. That is, the fit would be considered good if there is no significant divergence between the observed and expected data when the curve of the observed frequencies is super-imposed on the curve of expected frequencies.

We must remember, however, that even if proportions in the cells remain unchanged the chi-square value varies directly with the total number of cases (N). If we double the number of cases, the chi-square value gets doubled; if we triple the number of cases we also triple chi-square and so on.

**The implications of this fact can be illustrated by an example given below: **

In the present example, the chi-square value is 3.15. On this basis we would naturally infer that the relationship is not a significant one.

**Now, suppose data had been collected on 500 cases with the following results: **

Chi-square value as computed from the figures, is now 6.30, which is double the value arrived at in the previous example. The value 6.30 is statistically significant. Had we expressed the results in terms of percentages there would not have been a difference in interpretation.

The above examples illustrate a very important point, viz., that chi- square is directly proportional to N. Hence, we would need a measure which is not affected merely by a change in the number of cases. The measure phi (ǿ) affords this facility, i.e., the property we desire in our measure. This measure is simply a ratio between the chi-square value and the numerical total of the cases studied.

**The measure phi (ø) is defined as:**

Ø = √x^{2 }/n

that is, the square root of chi-square divided by the number of cases.

**Thus, by applying this formula to the two above cited examples we get, in the first case: **

Thus, the measure ø unlike the chi-square, gives the same result when the proportions in the comparable cells are identical.

G. Udny Yule has proposed yet another coefficient of association usually designated as “Q” (more commonly known as Yule’s Q) which measures association in? x 2 table. The coefficient of association (Q) is obtained by computing the ratio between the difference and the sum of the cross products of the diagonal cells, if cells of the 2×2 table are designated as in the following table:

ac- bc/ad + be

where a, b, c, and d refer to the cell-frequencies.

The coefficient of association Q varies between minus one and plus one (+1) as be is less than or greater than ad. Q attains its limits of +1 whenever any one of the cells is zero, i.e., the association is complete (the correlation is perfect). Q is zero when the variables are independent (that is, when there is no association), i.e., when ad. = be and. Q = 0.

**Application of the above formula is illustrated in the following example: **

**Let us compute Yule’s Coefficient of Association between marital status and performance in examination on the basis of the data presented in the following table: **

**Substituting the above values in the Yule’s formula: **

Thus, there is a slight negative association between marital status and performance in examination.

We may look at the problem from another point of view also.

The percentage of married students who failed is = 60 ×100/150 =40.

The percentage of unmarried students who failed is, =100×100/350 =28.57 (Approx.)

Thus, 40 per cent of the married students and nearly 29 per cent of the unmarried students failed in the examination. Hence the poor performance of students may be attributed to marital status.

Causal inferences can be very securely established in an experimental situations. We have considered this issue when dealing with experimental designs. In social sciences, it is very difficult to set up an experiment, so most of the studies are non- experimental ones. Analytical procedures have, however, been devised to draw inferences about causal relationships in non-experimental studies.

In as much as most social researches involve a study of the samples drawn from the ‘population’ and seek to draw generalizations to this ‘population’, it is necessary, in the interest of science, to know the extent to which generalizations thus drawn are justified.

Suppose, in a study on samples of male and female students our results show significant differences between the two samples in terms of the number of hours they devote to studies.

We may ask whether the differences that are observed reflect the true differences between the male and female students or whether the two ‘populations’ of students are in fact alike in terms of the hours they devote to studies but the samples drawn from these ‘populations’ for the study might have differed to this extent by ‘chance.’

A number of statistical procedures have been designed to enable us to answer such a question in terms of the statements of probability.

When we are comparing samples or studying the difference between experimental and control groups, we normally wish to test some hypothesis about the nature of the true difference between the ‘populations’ supposed to be represented by the samples under study.

In social sciences, we are usually concerned with relatively crude hypothesis (for example, the female students devote more time to their studies than the male students do).

We are not usually in a position to consider more specific or exact hypothesis (e.g., which specify in exact terms the difference between the two ‘populations’). Suppose, our data show that the sample of female students devotes on an average four hours to studies whereas the sample of male students, only two hours.

Clearly, the findings of our samples are in tune with the hypothesis, i.e., female students devote more time to their studies than their male counterparts. But we must constantly bear in mind the possibility that the findings based on our samples may not be exactly the same as the findings we might have obtained had we studied two ‘populations’ in toto.

Now, we want to estimate whether we would still have observed more time spent on studies among the female students, had we studied the total ‘population’. Such an estimate is possible if we test the ‘null hypothesis.’

The ‘null hypothesis’ states that the ‘populations’ do not differ in terms of characteristics under study. In this case, a ‘null hypothesis’ would state that in the larger ‘population’ of students as a whole, subgroups of the female and male students do not differ in respect of the time they devote to their studies.

Various statistical techniques called the tests of significance, have been devised which help us estimate the likelihood that our two samples might have differed to the extent they do, by chance, even if there is actually no difference between the two corresponding ‘populations’ of male and female students in respect of time devoted to studies.

Among the various methods of testing significance, the decision as to which method will be appropriate for a particular study depends on the nature of the measurements used and distribution of the characteristics (e.g., hours of study, number of children, salary-expectations etc.).

Most of these tests of significance assume that the measurements constitute interval scale and that the distribution of the characteristic approximates a normal curve. In social research, these assumptions seldom correspond to reality. Recent statistical developments have, however, come out with some kind of a solution to this, in the form of non-parametric tests that do not rest on these assumptions.

We should try to understand at this point the reason why the ‘null hypothesis’ should be tested when our actual interest is in testing a hypothesis (alternative hypothesis, as it is called) which states that there is a difference between the two ‘populations’ represented by the samples.

The reason is easy to appreciate. Since we do not know the true picture in the ‘population’ the best we can do is to make inferences about it on the basis of our sample-finding.

**If we are comparing two samples, there are, of course, two possibilities: **

(1) Either, the populations represented by the sample are alike or

(2) They are different.

Our samples from two ‘populations’ are different in respect of some attributes; hours devoted to studies in our example. Clearly, this could happen if the two ‘populations’ which the samples represent do in fact differ in respect of the said attribute.

This, however, does not constitute a definitive evidence that these ‘populations’ differ, since there is always the possibility that the samples do not exactly correspond to the ‘populations’ they purport to represent.

We must therefore allow room for the possibility that the element of chance which is involved in the selection of a sample may have given us samples which differ from each other although the two ‘populations’ from which they are drawn do not in fact differ.

**The question we may want to ask, therefore, is: **

“Could we have possibly got samples differing from each other to the extent they do, even if the ‘populations’ from which they are drawn did not differ?” This precisely is the question which a ‘null hypothesis’ answers.

The ‘null hypothesis’ helps us to estimate what the chances are that the two samples differing to this extent would have been drawn from two ‘populations’ that are in fact alike: 5 in 100? 1 in 100? or whatever.

If the statistical test of significance suggests that it is improbable that two samples differing to this extent could have been drawn from ‘populations’ that are in fact similar, we may conclude that the two ‘populations’ probably differ from each other.

A point to bear in mind here is that all statistical tests of significance and thus all generalizations from the samples to the populations rest on the assumption that the samples are not selected in a manner that bias could have entered into the process of drawing the samples.

In other words, the assumption is that the sample we have selected has been drawn in such a manner that all cases or items in the ‘population’ had an equal or specifiable chance of being included in the sample.

If this assumption is not justified, the tests of significance become meaningless and inapplicable. In other words, the tests of significance apply only when the probability principle had been employed in selecting the sample.

To return to our illustration, suppose, our findings show no difference between the two samples: which means that both male and female students in our sample are found to devote equal time to their studies.

Can we then say that the two ‘populations’ of male and female students are similar in terms of this attribute? Of course, we cannot say this with any certainty since there is a possibility that the samples may be alike when the populations actually differ.

But to go back to the case where the two samples differ, we can affirm that the two populations they represent probably differ if we can reject the ‘null hypothesis’; that is, if we can show that the difference between the two samples is unlikely to appear if the above ‘populations’ did not differ.

But again, there is some chance that we may be wrong in rejecting the ‘null hypothesis’ since it is in the nature of probability that even highly improbable events may sometimes take place.

There is another side to it, too. Just as we may be wrong in rejecting the ‘null hypothesis,’ it is also likely that we may be wrong in accepting the ‘null hypothesis.’ That is, even if our statistical test of significance indicates that sample differences might easily have arisen by chance even though the ‘populations’ are similar, it may nevertheless be true that the ‘populations’ do in fact differ.

**In other words, we are always faced with the risk of making any one of the two types of error: **

(1) We may reject the ‘null hypothesis’ when in fact it is true,

(2) We may accept the ‘null hypothesis’ when in fact it is false.

The first type of error, we may call the Type I error. This consists in inferring that the two ‘populations’ differ when in fact they are alike.

The second type of error may be called the Type II error. This consists in inferring that the two ‘populations’ are alike when in fact they differ.

The risk of making the Type I error is determined by the level of significance we are prepared to accept in our statistical testing, e.g., 0.05, 0.01, 0.001, etc. (that is, 5 in 100, 1 in 100 and 1 in 1000 . Thus, if we decide, for instance, that the populations truly differ whenever a test of significance shows that the difference between the two samples would be expected to occur by chance not more than 5 times in 100.

This means that if the two ‘populations’ represented by the sample were in fact similar (in terms of a certain attribute), then we are accepting 5 chances in 100 that we will be wrong in rejecting the ‘null hypothesis.’ We may, of course, minimize the risk of making Type I error by making our criterion for rejecting the null hypothesis, more strict and tight.

We may, for example, decide the level of significance at 0.01, i.e., we would reject the ‘null hypothesis’ only if the test shows that the difference in the two ‘samples’ might have appeared by chance only once in a hundred.

In essence, what we are saying is that we will reject the ‘null hypothesis’ if the test shows that out of a hundred samples of a designated size selected from the respective ‘populations’ by employing the probability principle, only one sample will show difference in terms of the attributes to the extent this is seen in the two samples under study.

The criterion for rejecting the ‘null hypothesis’ can be made even more strict by further elevating the level of significance. But the difficulty here is, that the errors of Type I and Type II happen to be so related to each other that the more we protect ourselves against making a Type I error, the more vulnerable we are to make a Type II error.

Having determined the extent of risk of type I error we are willing to run, the only way of reducing the possibility of Type II error is to take larger samples and use statistical tests that make the maximum use of available relevant information.

The situation with respect to the Type II error can be illustrated in a very precise way by means of an **“opening characteristic curve.”** The behaviour of this curve depends on how large the sample is. The larger the sample, the less probable it is that we will accept a hypothesis which suggests a state of affairs that is extremely far off from the state of reality.

In so far as the relationship between the Type I and Type II errors is inverse, it is necessary to strike a reasonable balance between the two types of risk.

In social sciences, it has almost become an established practice or convention to reject the ‘null hypothesis’ when the test indicates that the difference between the samples would not occur by chance more than 5 times out of 100. But the conventions are useful when there is no other reasonable guide.

The decision as to how the balance between the two kinds of error should be struck must be made by the researcher. In some instances, it is more important to be certain of rejecting a hypothesis when it is false than fail to accept it when it is true. In other cases the reverse may be true.

For example, in certain countries, it is considered more important to reject a hypothesis of guilt when it is false than to fail to accept this hypothesis when it is true, i.e., a person is considered not guilty so long as there is a reasonable doubt about his guilt. In certain other countries, a person charged with a crime is considered guilty until such a time as he has demonstrated his lack of guilt.

In much research, of course, there is no clear basis for deciding whether a Type I or Type II error would be more costly and so the investigator makes use of the conventional level in determining statistical significance. But, there may be some studies in which one type of error would clearly be more costly and harmful than the other.

Suppose, in an organization it has been suggested that a new method of division of labour would be more effective and suppose also that this method would require a lot of expense.

If an experiment constituted of two groups of personnel — one operating as experimental group and the other, as control group — is set up to test whether the new method is really beneficial for the organizational goals and if it is anticipated that the new method would entail a lot of expenses, the organization would not wish to adopt it unless there was considerable assurance of its superiority.

In other words, it would be expensive to make a Type 1 error, i.e., conclude that the new method is better when it is not so in fact.

If the new method entailed expenses which were about the same as the old method, then type II error would be undesirable and more damaging since it may lead to failure on the part of the management to adopt the new method when it is in fact superior and as such has long-range benefits in store for the organization.

Any generalizations from the sample to the ‘population’ is simply a statement of statistical probability. Let us say, we have decided to work with a 0.05 level of significance. This means that we shall reject the ‘null hypothesis’ only if the sample-difference of the magnitude that we have observed can be expected to occur by chance not more than 5 times in 100.

Of course, we will accept the ‘null hypothesis’ if such a difference can be expected to occur by chance more than 5 times out of 100. Now the question is: Does our finding represent one of those 5 times when such a difference might have appeared by chance?

This question cannot be definitively answered on the basis of an isolated finding. It may, however, be possible for us to say something about this when we examine the patterns within our findings.

Suppose we are interested in testing the effects of a film on attitudes toward a particular governmental programme, say family planning. We have, let us say, taken full care to keep the desired conditions for experimentation at the maximum.

Now suppose that we use as one measure of attitudes toward the programme, only one item, viz., the attitude toward spacing children and find that those who saw the film are more favorably inclined toward this issue than those who did not see the film.

Now suppose, the statistical test shows that the difference would not have appeared by chance due to random sampling fluctuations more than once in twenty. Logically, it also means that it might have appeared by chance once in twenty (or 5 times in 100). As we have pointed out, we have no definite way of knowing whether our sample is one among the five in 100. Now, what best can we do?

Let us say, we have asked 40 different questions to respondents, which are reasonable indicators of the attitude toward the family welfare governmental programme. If we are using a confidence level of 5% and if we asked 100 questions we might expect to find statistically significant differences attributable to chance on 5 of them.

Thus, out of our 40 questions on various items, we may expect to find statistically significant differences on 2 of them. But, suppose we actually find that on 25 out of 40 questions on those who saw the film had more favourable attitudes compared to those who did not see the film.

We may, this being the case, feel much safer in concluding that there is a true difference in attitudes (even though the statistical test indicates that the difference might have arisen by chance on each question 5 times in 100).

Now let us suppose that out of the 40 questions, responses to only one, i.e., about the spacing of children, showed a statistically significant difference between the two groups those exposed to film and those not). This difference might as well have occurred by chance.

On the other hand, it may be that the content of the film actually did influence opinions on this point but not on any other (such as that relating to sterility operations). But unless our hypothesis has specifically predicted in advance that the film would be more likely to affect the attitudes toward spacing of children than the attitudes toward any of the other 39 questions, we are not justified in making this interpretation.

Such an interpretation, i.e., one invoked to explain the findings after they surface, is known as the ‘post-factum’ interpretation, because this involves explanations provided to justify the findings whatever they are. It depends upon the ingenuity of the researcher, as to what explanation he can invent to justify this findings. He can, therefore, justify even the opposite findings.

Merton has very lucidly pointed out that the post-factum interpretations are designed to “explain” observations. The method of post-factum explanation is completely flexible. If the researcher finds that the unemployed tend to read fewer books than they did previously, this may be “explained” by the hypothesis that anxiety resulting from unemployment affects concentration and so reading becomes difficult.

If, however, it is observed that the unemployed read more books than previously (when in employment), a new post-factum explanation can be invoked; the explanation being that the unemployed have more leisure and, therefore, they read more books.

The critical test on ‘an obtained relationship (among variables) is not the post-factum rationales and explanation for it; it is rather the ability to predict it or to predict other relationships on the basis of it. Thus, our previously unpredicted finding of a difference in attitudes toward spacing of children, even though statistically significant, cannot be considered as established by the study we have carried out.

Since, statistical statements are statements of probability, we can never totally rely on the statistical evidence alone for deciding whether or not we will accept a hypothesis as true.

Confidence in the interpretation of a research result requires not only statistical confidence in the reliability of the finding (i.e., that the differences are not likely to have occurred by chance) but in addition, some evidence about the validity of presuppositions of the research.

This evidence is necessarily indirect. It comes from the congruence of the given research findings with other knowledge which has withstood the test of time and hence about which, there is considerable assurance.

Even in the most rigorously controlled investigation, the establishment of confidence in the interpretation of one’s results or in the imputation of causal relationships requires replication of research and the relating of the findings to those of other studies.

It is necessary to note that even when statistical tests and the findings of a number of studies suggest that there is indeed a consistent difference between two groups or consistent relationship between two variables, this still does not constitute the evidence of the reason for the relationship.

If we want to draw causal inferences (e.g., X produces Y), we must meet assumptions over and above those required for establishing the existence of a relationship. It is also worthy of note that a result is not socially or psychologically significant just because it is statistically significant. Many statistically significant differences may be trivial in practical social parlance.

For example, an average difference of less than one I.Q. point between the urban and rural people may be significant statistically, but not so in the practical day-to-day life. Contrariwise, there are cases where a small but reliable difference has great practical significance.

In a large-scale survey, for example, a difference of half or one per cent may represent hundreds of thousands of people and awareness of the difference may be important for significant policy decisions. Therefore, the researcher besides being concerned with the statistical significance of his findings must also be concerned with their social and psychological meanings.

**Inferring Causal Relationships: **

Owing to obvious difficulties, such rigid experimental designs can rarely be worked out in social scientific investigations. Most of the inquiries in social sciences are non-experimental in character.

In such studies, there are certain empirical obstacles in the way of determining whether or not a relation between variables is causal. It has been repeatedly mentioned that one of the most difficult tasks in the analysis of social behaviour data is the establishment of cause and effect relationships.

A problematic situation owes its origin and the process of becoming, not just to one factor but to a complex of a variety of factors and sequences.

The process of disentangling these elements poses a major challenge to the sociological imagination and puts to test the skill of researchers. It is dangerous to follow a ‘one- track’ explanation which leads to the cause. It is imperative to look for a whole battery of causal factors which generally play a significant role in bringing about complex social situations.

As Karl Pearson aptly observes, **“no phenomenon or stage in sequence has only one cause; all antecedent stages are successive causes; when we scientifically states causes, we are really describing the successive stages of a routine of experience.” **

Yule and Kendall have recognized the fact that statistics “must accept for analysis, data subject to the influence of a host of causes and must try to discover from the data themselves which causes are the important ones and how much of the observed effect is due to the operation of each.”^{ }

Paul Lazarsfeld has traced the phases involved in the technique he calls ‘discerning.’ He advocates its use in determining causal relations between variables. Lazarsfeld lays down this procedure:

**(a) ****Verifying an alleged occurrence as under: **

In order to verify this occurrence, it is necessary to ascertain if the person has actually experienced the alleged situations. If so, how does the occurrence manifest itself and under what conditions, in his immediate life?

What reasons are advanced for the belief that there is a specific interconnection between two variables, e.g., loss of employment and loss of authority? How correct is the person’s reasoning in this particular instance?

(b) Attempting to discover whether the alleged condition is consistent with objective facts of the past life of this person.

(c)Testing all possible explanation for the observed condition.

(d) Ruling out those explanations which are not in accord with the pattern of happenings.

It is quite understandable that most difficulties or obstacles to establishing causal relationships afflict non-experimental studies most sharply. In non-experimental studies where the interest is in establishing causal relationships among two variables, the investigator must find substitutes for safeguards that are patently built into the experimental studies.

Many of these safeguards enter at the time of planning data- collection, in the form of providing for the gathering of information about a number of variables that might well be the alternative conditions for producing the hypothesized effect.

By introducing such additional variables into the analysis, the researcher approximates some of the controls that are inherent in experiments. Nevertheless, the drawing of inferences of causality does always remain somewhat hazardous in non-experimental studies.

We shall now discuss some of the problems and the strategies to overcome them, relating to drawing inferences about causality in non-experimental studies. If a non-experimental study points to a relationship or association between two variables, say X and Y, and if the research interest is in causal relationships rather than in the simple fact of association among variables, the analysis has taken only its first step.

The researcher must further consider (besides association between X and Y) whether Y (effect) might have occurred before X (the hypothesized cause), in which case Y cannot be the effect of X.

In addition to this consideration, the researcher must ponder over the issue whether factors other than X (the hypothesized cause) may have produced Y (the hypothesized effect). This is generally taken care of by introducing additional variables into the analysis and examining how the relation between X and Y is affected by these further variables.

If the relationship between X and Y persists even when other presumably effective and possibly alternative variables are introduced, the hypothesis that X is the cause of Y remains tenable.

For example, if the relation between eating a particular seasonal fruit (X) and cold (Y) does not change even when other variables such as age, temperature, state of digestion, etc., are introduced into the analysis, we may accept the hypothesis that X leads to Y as tenable.

But it is possible in no small number of cases that the introduction of other additional variables may change the relationship between X and Y. It may reduce to totally eliminate the relationship between X and Y or it may enhance the relationship in one group and reduce it in another.

If the relationship between X (eating of seasonal fruit) and Y (cold) is enhanced in a sub-group characterized by Z (bad state of digestion) and reduced in sub-group not characterized by Z (normal state of digestion), we may conclude that Z is the contingent condition for the relationship between X and Y.

This means, in other words, that we have been able to specify condition (Z) under which the relation between X and Y holds. Now if introduction of Z in the analysis reduces or totally eliminates the relationship between X and Y, we shall be safe in concluding either that X is not a producer of Y, that is, the relation between X and Y is ‘spurious’ or that we have traced the process by which X leads to Y (i.e., through Z).

Let us turn to consider the situation in which we can legitimately conclude that the relation between X and Y is spurious.

An apparent relationship between two variables X and Y is said to be spurious if their concomitant variation stems not from a connection between them but from the fact that each of them (X and Y) is related to some third variable (Z) or a combination of variables that does not serve as a link in the process by which X leads to Y.

**The situation characterizing spurious relationship may be diagrammed as under: **

The objective here is to determine the cause of Y, the dependent variable (let us say, the monetary expectation by college graduates). The relationship (broken line) between X the independent variable (let us say, the grades obtained by students) and the monetary expectation of graduates (Y) has been observed in the course of the analysis of data.

Another variable (Z) is introduced to see how the relation between X and Y behaves with the entry of this third factor. Z is the third factor (let us say, the income-level of the parents of students). We find that the introduction of this factor reduces the relationship between X and Y.

That is, it is found that the relation between higher grade in the examination and higher monetary expectations does not hold itself up, but is considerably reduced when we introduce the third variable, i.e., the level of parents’ income.

Such an introduction of Z brings to light the fact that not X but Z may probably be a determining factor of Y. So the relationship between X and Y (shown in the diagram by a dotted line) is a spurious one, whereas the relation between Z and Y is a real one. Let us illustrate this with the help of hypothetical data.

Suppose, in the course of the analysis of data in a study, it was seen that there is a significant correlation between the grades or divisions (I, II, III) that students secured in the examination and the salary they expect for a job that they might be appointed to.

It was seen, for instance, that generally the first divisioners among students expected a higher remuneration compared to the second divisioners and the second divisioners expected more compared to the third divisioners.

**The following table illustrates the hypothetical state of affairs:**

It is clearly seen from the table that there is a basis for hypothesizing that the grades of the students determine their expectations about salaries. Now, let us suppose that the researcher somehow hits upon the idea that the income-level of the parents (X) could be one of the important variables determining or influencing the students’ expectations about salaries (Y). Thus, Z is introduced into the analysis.

**Suppose, the following table represents the relationship among the variables: **

**Note:**

H.M.L in the horizontal row, dividing each category of the students’ grades, stand respectively for high parental level of income, moderate parental level of income and low parental level of income. The above table clearly shows that the relation between X and Y has become less significant compared to the relation between Z and Y. ‘

To get a clearer picture, let us see the following table (a version of Table B omitting the categories of X) showing the relationship between Z and, i.e., parental income level and students’ monetary expectations:

We can very clearly see from the table that, irrespective of their grades, the students’ monetary expectations are very strongly affected by the parental levels of income (Z).

We see that an overwhelming number of students (i.e., 91.5%) having high monetary expectations are from the high parental income group, 92% having moderate monetary expectations are from moderate parental income group and lastly, 97% having low monetary expectations are from the low parental income group.

Comparing this picture with the picture represented by Table A, we may say that the relation between X and Y is spurious, that is, the grade of the students did not primarily determine the level of the monetary expectations of the students.

It is noted in Table A that students getting a higher grade show a significant tendency toward higher monetary expectations whereas the lower grade students have a very marked concentration in the lower monetary expectation bracket.

But when we introduce the third variable of parental income, the emerging picture becomes clear enough to warrant the conclusion that the real factor responsible differential levels of monetary expectations is the level of parental income.

In Table C, we see a very strong and formidable concentration of cases of students corresponding to the three under mentioned combinations, viz., of higher monetary expectations and higher parental income, of moderate monetary expectations and moderate parental income and of lower monetary expectations and lower parental income, i.e., 5%, 92.1% and 1% respectively.

Tracing the Process Involved and a Relationship Among Variables: As was stated earlier, if a third factor Z reduces or eliminates the relationship between the independent variable X and the dependent variable Y, we may conclude either that the relationship between X and Y is spurious, or that we have been able to trace the process by which X leads to Y.

We shall now consider the circumstances that would warrant the conclusion that the process of relationship between X and Y has been traced through a third factor Z.

Suppose, in a study the investigators found that smaller communities had a higher average intimacy score, the intimacy score being a measure of the intimacy of association between members of a community arrived at by using an intimacy scale.

Suppose, they also found that the middle-sized communities had a lesser intimacy score compared to the small-sized communities and big-sized communities had the least average intimacy score. Such a finding suggests that the size of the community determines the intimacy of association among members of the community.

In other words, the observations warrant the conclusion that the members living in a small-sized community have a greater intimacy of association, whereas the big-sized communities are characterized by a lesser intimacy of association among the members.

**The following table presents the hypothetical data: **

In the second column of the table, samples corresponding to each of the communities have been shown.

In the second column of the table, samples corresponding to each of the communities have been shown. In column 3, the average intimacy scores corresponding to types of communities calculated on the basis of the responses given to certain items on a scale relating to the day-to-day associations among members have been shown.

It is seen from the table that the average intimacy scores vary inversely with the size of the community, i.e., smaller the size, the greater the intimacy score and conversely, larger the size, the lower the intimacy score.

Now suppose, the investigators got the idea that the three types of communities would differ in terms of opportunities they offer for interaction among members, in as much as the living arrangements, residential patterning, commonly-shared utilities etc., would promote such association.

Thus, the investigators would introduce the third factor into analysis of the interaction-potential, i.e., the extent to which the circumstances persons live in are likely to provide opportunities for interaction among themselves.

In order to check the hypothesis that it was largely through differences in residential patterning, living arrangements, commonly shared amenities etc., that the three types of communities produced differences in interaction among members of a community, the investigators would consider the size of community and interaction-potential jointly in relation to the average intimacy score.

The infraction-potential is thus the third variable Z introduced into the analysis. The interaction-potential is classified, let us say, into a low interaction-potential (b) medium interaction potential, and (c) high interaction- potential.

**The following table represents the hypothetical data: **

Reading across the rows in the table, we see that the interaction-potential is strongly related to the intimacy score of the community members, whatever the size of the community.

That is, whether we consider the row for small-sized communities, for the middle-sized communities, or for the big-sized communities, there is in each case an increase in the average intimacy score with an increase in interaction-potential. Moreover, reading the entries across the rows, it becomes clear that the size of the community and the interaction-potential bear a significant correlation.

For example, approximately two-thirds of respondents in a small-sized community are living under conditions of high interaction-potential; we also find that a much smaller proportion of the moderate-sized community residents are living under high interaction-potential conditions and a very small proportion of the big-sized community residents under high interaction-potential conditions.

Now, we read the intimacy scores down the columns only to find that the relationship between the type of community and intimacy of association has been considerably reduced. In fact, for people living under high interaction potential conditions, no definite relationship between the size of the community and the intimacy score obtains.

From this set of relationships, the investigators may conclude that the inverse relationship between the size of the community and the intimacy score does hold good, but that one of the major ways in which a particular type of community encourages intimacy among its members is by offering opportunities that increase the rate of interaction amongst them.

In other words, the small-sized communities are characterized by a higher average intimacy score because their small size provides a setting for many opportunities for high degree of interaction among members. Big-sized communities, on the other hand, are characterized by a relatively lower intimacy score.

But the lower intimacy score is attributable not to the size of the community per se but to the fact that a big-sized community cannot offer opportunities for higher interaction among members as the small-sized communities do.

Hence, the investigators rather than concluding that the relationship between the size of the community and the average intimacy score among members is spurious, might conclude that they have been able to trace the process by which X {i.e., the type of community) influences Y (the intimacy score).

The former warranted the conclusion that the relation between the variables X and Y was spurious and the latter the conclusion that the process from X to Y may be traced through Z (X to Z to Y). In both cases, the introduction of a third variable Z reduced or eliminated the relationship between them (X and Y).

One difference may, however, be noted. In the first example, the variable Z (i.e., income level of parents) was clearly prior in time to the other two variables (grade of students in the examination and monetary expectations of students).

In the second example, the third variable Z (interaction-potential afforded by the communities) did not occur before the assumed causal variable (size of community). It was concurrent with it and might be thought of as starting after it.

The time-sequence of the variables, thus, is an important consideration in deciding whether an apparent causal relationship is spurious. That is, if the third variable Z, which removes or eliminates the relationship between the originally related variables X and Y, we conclude usually that the apparent causal relationship between variables X and Y is spurious.

But if the third variable Z is known or assumed to have occurred at the same times as X or after X, it may be in order to conclude that the process by which X leads to Y has been traced.Thus, to have certain measure of confidence in causal relationship inferred from studies that are non-experimental in character, it is necessary to subject them to the critical test of eliminating the other possibly relevant variables.

For this reason, it is important to collect in the course of study, data on such possibly influential variables other than those with which the hypothesis of the study are centrally concerned.

It was stated earlier that the introduction of a third variable into the analysis may have the effect of intensifying the relationship within one sub-group and of reducing the same in another sub-group. If such be the case, we say that we have specified a condition (Z) under which the relationship between X and Y holds.

Let us now illustrate the process of specification. Suppose, in a community study, we happen to identify a relationship between income and educational level.

**This is shown in the table given below: **

We see in the table that the relationship between education and income is a fairly marked one. Higher the education, generally, higher the percentage of cases earning a yearly income of Rs.5,000/- and above. However, we may decide that the relationship requires further specification.

That is, we may wish to know more about the conditions under which this relationship obtains. Suppose, the thought strikes us that the fact of the respondents living in urban-industrial community might positively affect the advantages of education for remunerative employment and hence its reflection in income.

On this assumption, we introduce the third factor Z, i.e., those respondents who live in the urban industrial community and those who live in the rural non-industrial community, into the analysis and see how it affects the initial relationship between X and Y (i.e., education and income).

**Suppose we get a picture as shown in the following table: **

We can see clearly that the Table B reflects a very different relationship between income and education for the people living in the rural-non-industrial community as compared to the one for those living in the urban-industrial community. We see that for those living in the industrial cities, the relationship between education and income is somewhat higher than the original relationship.

But, for those living in the rural non- industrial communities the relationship in the above table is considerably lower than the initial relationship.

Thus, the introduction of the third factor and the break-down of the original relationship on the basis of the third factor (Z) has helped to specify a condition under which the relationship between X and Y is more pronounced as also the condition under which the relation is less pronounced.

Similarly suppose, we find in the course of a study that people who belong to the higher income category have generally lesser number of children compared to those in the lower income category. Suppose, we feel (on the basis of a theoretic orientation) that the factor of city-dwelling could be important in affecting the relationship.

Introducing this factor, suppose, we find that the original relationship between level of income and number of children becomes more pronounced in the city and that it becomes less pronounced among the rural people, than we have identified a condition Z (i.e., city- dwelling) under which the relation becomes decisively enhanced or pronounced.

**Interpreting the Findings of a Study: **

Thus far, we have concerned ourselves mainly with the procedures that together comprise, what we call customarily, the analysis of data. The researcher’s task, however, is incomplete if he stops by presenting his findings in the form of empirical generalizations which he is able to arrive at through the analysis of data.

A researcher who, for example, winds up his research exercise just by stating that **“the unmarried people have a higher incidence of suicide as compared to the married people”** is hardly fulfilling his overall obligation to science, though the empirical generalization he has set forth does have some value by itself.

The researcher in the larger interest of science must also seek to show that his observation points to certain under-laying relations and processes which are initially hidden to the eye. In other words, the researcher must show that his observation has a meaning, much broader and deeper, than the one it appears to have on the surface level.

To return to our example of suicide, the researcher should be able to show that his observation that “the unmarried people are characterized by suicide” reflects, in fact, the deeper relationship between social cohesion and rate of suicide (Durkheim’s theory).

Once the researcher is able to expose the relations and processes that underlie his concrete findings he can establish abstract relationships between his findings and various others.

In essence then, the researcher’s work goes well beyond the collection and analysis of data. His task extends to interpreting the findings of his study. It is through interpretation that the researcher can understand the real significance of his findings, i.e., he can appreciate why the findings are what they are.

As was stated earlier, interpretation is the search for broader and more abstract meanings of the research findings. This search involves viewing the research findings in the light of other established knowledge, a theory or a principle. This search has two major aspects.

The first aspect involves the effort to establish continuity in research through linking the results of a given study with those of another. It is through interpretation that the researcher can unravel or comprehend the abstract principle beneath the concrete empirical observations.

This abstract common denominator having been discerned, the researcher can easily proceed to link his findings up with those of other studies conducted in diverse settings, diverse in matters of detail but reflecting the same abstract principle at the level of findings.

Needless to say that the researcher can on the basis of the recognition of the abstract theoretic principle underlying his finding, make various predictions about the concrete world of events quite unrelated seemingly to the area of his findings. Thus, fresh inquiries may be triggered off to test predictions and understandably, such studies would have a relationship with the researcher’s initial study.

In a somewhat different sense, interpretation is necessarily involved in the transition from exploratory to experimental research. The interpretation of the findings of the former category of researches often leads to hypotheses for the latter.

Since, an exploratory study does not have a hypothesis to start with, the findings or conclusions of such a study have to be interpreted on a ‘post-factum’ interpretation is often a hazardous game fraught with dangerous implications. Such an interpretation involves a search for a godfather in the nature of some theory or principle that would adopt (i.e., explain) the findings of the study.

This quest often turns out to be an exercise on the part of the researcher to justify his findings by locating some suitable theory to fit his findings. As a result quite so often contradictory conclusions may find their ‘godfathers’ in diverse theories.

This aspect of post-factum interpretation, comprising attempts at rationalizing the research findings, should be clearly borne in mind when proceeding with it. On occasions there is, however, no other alternative to it.

Secondly, interpretation leads to the establishment of explanatory concepts. As has been pointed out, interpretation of findings involves efforts to explain why the observations or findings are, what they are. In accomplishing this task, theory assumes central importance.

It is a sensitizer and a guide to the underlying factors and processes (explanatory bases) beneath the findings. Underneath the researcher’s observations in the course of a study, lies a set of factors and processes which might explain his observations of the empirical world. Theoretical interpretation uncovers these factors.

The researcher’s task is to explain the relations he has observed in the course of his study, by exposing the underlying processes which afford him a deeper understanding of these relations and point to the role of certain basic factors operating in the problem area of his study.

Thus, interpretation serves a twofold purpose. First, it gives an understanding of the general factors that seem to explain what has been observed in the course of a study and secondly, it provides a theoretical conception which can serve in turn as a guide for further research.

It is in this manner that science comes to cumulatively disengage more successfully the basic, processes which shape the portion of the empirical world with which a researcher is concerned.

Interpretation is so inextricably intertwined with analysis that it should more properly be conceived of as a special aspect of analysis rather than a separate or distinct operation. In closing, we are tempted to quote Prof. C. Wright Mills who has stated the very essence of what all is involved in the analysis (involving interpretation) of data.

Says Mills, “So you will discover and describe, setting up types for the ordering what you have found out, focusing and organizing experience by distinguishing items by name. This search for order will cause you to seek patterns and trends and find relations that may be typical and causal. You will search in short, for the meaning of what you have come upon or what may be interpreted as a visible token of something that seems involved in whatever you are trying to understand; you will pare it down to essentials; then carefully and systematically you will relate these to one another in order to form a sort of working model….”

**“But always among all details, you will be searching for indicators that might point to the main drift, to the underlying forms and tendencies of the range of society in its particular period of time.”** After a piece of research is terminated, the statement that raises an array of new questions and problems may be made.

Some of the new questions constitute the groundwork for new research undertakings and formulation of new theories which will either modify or replace old ones. This is indeed, what research means. It serves to open new and more wider avenues of intellectual adventure and simulates the quest for more knowledge as well as greater wisdom in its use.