Lord’s paradox was first raised by Lord in an article on “Psychological Bulletin” in 1967. It reveals a contrast of two statisticians’ conclusions based on the same set of data. Here are the Lord’s four examples to illustrate his points.
Example1: “A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex differences in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and his weight the following June are recorded. ”(Lord,1967)
Example 2: “A group of underprivileged students is to be compared with a control group on freshman grade-point average than the control group. However, the underprivileged group started with a considerably lower mean aptitude score(x) than did the control group. Is the observed difference between groups on y attributable to initial differences on x? or shall we conclude that the two groups achieve differently even after allowing for initial differences in measured aptitude” (lord,1969)
Example 3 “suppose an agronomist is studying the yield of various varieties of corn. He plants 20 flower pots with seeds of a “white” variety. For simplicity of illustration. Suppose that he treat all 40 plants equally for several months, after which he finds that the white variety has yielded considerably more marketable grain than the black variety. However, it is a fact that black variety plants average only 6 feet high at flowering time: whereas white variety plants average 7 feet. He now asks the question, would the black variety process as much salable grain if conditions were adjusted so that it averaged 7 feet in height at flowering time?” ( Lord,1969)
Example 4: “consider the problem of evaluating federally funded special education programs. A group of disadvantaged children are pretested in September, then enrolled in a special program, and finally posttested in June. A control group of children are similarly pretested and posttested but not enrolled in the special program. Since the most disadvantaged children are selected for the special program, the control group will typically have higher pretest scores than the disadvantaged group”(Lord,1973)
Through analyzing four examples, it is easily found that the previous inconsistence on conditions of experimental group and control group make the statistical hypothesis difficult to balance them. Therefore, it is necessary to apply counterfactual thinking to modify the previous condition in theory. However, the question is whether it is available to such modifications.
Rubin(1982) investigated the Lord's paradox in the form of Rubin causal model. The first statistician assumes the causal effects in a form below.
（quoted from Holland, Paul W. Rubin, Donald B,1982)
E represents the experimental effect,
The first statistician finds that there is no difference between the concomitant variable and outcome variable for both females and males.
Considering the previously differences of experimental group and control group before the treatment and attempting to use covariance to control it , the second statistician computes the causal effect in such a way below:
(quoted from Holland, Paul W. Rubin, Donald B,1982)
If we convert the equation of causal effect into a regression form, we can get the equations below:
(quoted from Holland, Paul W. Rubin, Donald B,1982)
In the equation,
From the view of Rubin causal model, there is a concomitant variable (X) closely associated with the outcome variable (Y) in the example of Lord’s paradox. The concomitant variables produce the previously unequal conditions and affect the following outcome variables.
Rubin pointed out that there are some underlying assumptions in the statistical hypothesis the statistician made such as which is not testified appropriately. These assumptions have influenced the conclusions the statisticians reach
Rubin identify three types of studies as descriptive studies, uncontrolled causal studies and the controlled causal studies. The descriptive studies have no experimental manipulation. The uncontrolled causal studies have experimental manipulations without strictly controlling relevant factors. All possible factors have been sufficiently considered by experimenter in the controlled causal studies. The first statistician uses the unconditional descriptive statement that control group and experimental group are equal before treatment. The second statistician uses conditional statement which considers the previous differences. If both statisticians use descriptive statements, they are both right. However, when the descriptive statements are converted into causal statement, neither of them is right.
If I apply the Rubin and Holland’s analysis to the example of Lord’s paradox in the Powerpoint （http://lixiaoxu.googlepages.com/08Dec2006.ALL.G.ppt）, I can assume P represents the students in the class; t represents the course students receive; G represents the genders of students; X represents the degree of confidence before the course; Y represents the degree of the confidence after the course. Then we can see two statisticians make the different conclusions. Because the first assumes average confidence gains for males and females are equal, and the second assumes the male students and female students have equal confidence before course. We can see that two statisticians respectively make the untestified underlying assumptions before raise their null hypothesis. Those two counterfactual thinking both go against the real causal chains. It is doubtful to simply alter the causal factor without careful consideration. It seems to be a paradox for me to setting the experimental condition according the hypothesized causal relationship to explore the real causal relationships.
Lord’s paradox reflects the influence of differences of statistical hypothesis on the ultimate conclusion made. Can statistical hypothesis recognize and eliminate the previously existed unequality? In another words, descriptive statement and causal statements are two different language systems. The free translation between them seems not a certain thing. Since hypothesis test can only fix two contrast propositions, it is necessary for us to understand the limits of statistical language.
Reference
Holland, Paul W. Rubin, Donald B.(1982), On Lord's Paradox. Program Statistics Research
Lord, F. M.(1969) Statistical adjustments when comparing preexisting groups. Pwchological. Bulletin, 72, 336- 337
Lord, F. M.(1967) A paradox in the interpretation of group comparisons. Psychological Bulletin, 68 , 304-305
Example1: “A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex differences in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and his weight the following June are recorded. ”(Lord,1967)
Example 2: “A group of underprivileged students is to be compared with a control group on freshman grade-point average than the control group. However, the underprivileged group started with a considerably lower mean aptitude score(x) than did the control group. Is the observed difference between groups on y attributable to initial differences on x? or shall we conclude that the two groups achieve differently even after allowing for initial differences in measured aptitude” (lord,1969)
Example 3 “suppose an agronomist is studying the yield of various varieties of corn. He plants 20 flower pots with seeds of a “white” variety. For simplicity of illustration. Suppose that he treat all 40 plants equally for several months, after which he finds that the white variety has yielded considerably more marketable grain than the black variety. However, it is a fact that black variety plants average only 6 feet high at flowering time: whereas white variety plants average 7 feet. He now asks the question, would the black variety process as much salable grain if conditions were adjusted so that it averaged 7 feet in height at flowering time?” ( Lord,1969)
Example 4: “consider the problem of evaluating federally funded special education programs. A group of disadvantaged children are pretested in September, then enrolled in a special program, and finally posttested in June. A control group of children are similarly pretested and posttested but not enrolled in the special program. Since the most disadvantaged children are selected for the special program, the control group will typically have higher pretest scores than the disadvantaged group”(Lord,1973)
Through analyzing four examples, it is easily found that the previous inconsistence on conditions of experimental group and control group make the statistical hypothesis difficult to balance them. Therefore, it is necessary to apply counterfactual thinking to modify the previous condition in theory. However, the question is whether it is available to such modifications.
Rubin(1982) investigated the Lord's paradox in the form of Rubin causal model. The first statistician assumes the causal effects in a form below.
\inline \emph{$D_{i}=E(Y_{t}-X\lyxmathsym{\textSFxi}G=i),$ i=1,2} \inline \emph{$D=D_{1}-D_{2}$} （quoted from Holland, Paul W. Rubin, Donald B,1982)
E represents the experimental effect,
\inline \emph{$Y_{t}$ } represents the outcome variable in the experimental group, G represents the subpopulation indicator variable, X represents the concomitant variable. D represents causal effect.The first statistician finds that there is no difference between the concomitant variable and outcome variable for both females and males.
Considering the previously differences of experimental group and control group before the treatment and attempting to use covariance to control it , the second statistician computes the causal effect in such a way below:
\inline\emph{$D_{i}=E(Y_{t}-X\lyxmathsym{\textSFxi}G=i),$ i=1,2}\inline \emph{$D=D_{1}-D_{2}$}(quoted from Holland, Paul W. Rubin, Donald B,1982)
If we convert the equation of causal effect into a regression form, we can get the equations below:
\inline \emph{$D_{i}=E(Y_{t}-X\lyxmathsym{\textSFxi}X,G=i)=a_{i}+bX$ i=1,2,}\inline \emph{$D_{i}(x)=a_{i}+(b-1)x,$ i=1,2,}(quoted from Holland, Paul W. Rubin, Donald B,1982)
In the equation,
\inline $a_{i}$ represents the differences of previous conditions on experimental group and control group; b represents the influence of the causal effect in the experimental group. In Lord’s paradox, two statisticians have different underlying assumption on \inline $a_{i}$ and b. The first statistician assumes \inline $a_{i}$ is o and b is 1, while the second assumes b is same in two groups. Therefore, they make their conclusions according their different underlying assumptions.From the view of Rubin causal model, there is a concomitant variable (X) closely associated with the outcome variable (Y) in the example of Lord’s paradox. The concomitant variables produce the previously unequal conditions and affect the following outcome variables.
Rubin pointed out that there are some underlying assumptions in the statistical hypothesis the statistician made such as which is not testified appropriately. These assumptions have influenced the conclusions the statisticians reach
Rubin identify three types of studies as descriptive studies, uncontrolled causal studies and the controlled causal studies. The descriptive studies have no experimental manipulation. The uncontrolled causal studies have experimental manipulations without strictly controlling relevant factors. All possible factors have been sufficiently considered by experimenter in the controlled causal studies. The first statistician uses the unconditional descriptive statement that control group and experimental group are equal before treatment. The second statistician uses conditional statement which considers the previous differences. If both statisticians use descriptive statements, they are both right. However, when the descriptive statements are converted into causal statement, neither of them is right.
If I apply the Rubin and Holland’s analysis to the example of Lord’s paradox in the Powerpoint （http://lixiaoxu.googlepages.com/08Dec2006.ALL.G.ppt）, I can assume P represents the students in the class; t represents the course students receive; G represents the genders of students; X represents the degree of confidence before the course; Y represents the degree of the confidence after the course. Then we can see two statisticians make the different conclusions. Because the first assumes average confidence gains for males and females are equal, and the second assumes the male students and female students have equal confidence before course. We can see that two statisticians respectively make the untestified underlying assumptions before raise their null hypothesis. Those two counterfactual thinking both go against the real causal chains. It is doubtful to simply alter the causal factor without careful consideration. It seems to be a paradox for me to setting the experimental condition according the hypothesized causal relationship to explore the real causal relationships.
Lord’s paradox reflects the influence of differences of statistical hypothesis on the ultimate conclusion made. Can statistical hypothesis recognize and eliminate the previously existed unequality? In another words, descriptive statement and causal statements are two different language systems. The free translation between them seems not a certain thing. Since hypothesis test can only fix two contrast propositions, it is necessary for us to understand the limits of statistical language.
Reference
Holland, Paul W. Rubin, Donald B.(1982), On Lord's Paradox. Program Statistics Research
Lord, F. M.(1969) Statistical adjustments when comparing preexisting groups. Pwchological. Bulletin, 72, 336- 337
Lord, F. M.(1967) A paradox in the interpretation of group comparisons. Psychological Bulletin, 68 , 304-305
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