Lord’s paradox comparison of the approaches ADM study Lord, 1967 the two approaches change score approach: Y post − Y pre = β 1 + β 2 W + ǫ regressor variable approach Y post = β 1 + β 2 W + β 3 Y pre + ǫ still regressor variable approach Y post − Y pre = β 1 + β 2 W + ( β 3 − 1) Y pre + ǫ Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 the two approaches change score approach: Y post − Y pre = β 1 + β 2 W + ǫ regressor variable approach Y post = β 1 + β 2 W + β 3 Y pre + ǫ still regressor variable approach Y post − Y pre = β 1 + β 2 W + ( β 3 − 1) Y pre + ǫ Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study Lord, 1967 in other disciplines and traditions causal SEM framework statistician 1: total effect statistician 2: direct effect (adjusting for pretest) econometrics statistician 1: FD approach (same as FE estimator when two time-points) (also: DID) statistician 2: lagged dependent variable approach experimental design statistician 1: ANOVA (RANOVA: repeated measures ANOVA) statistician 2: ANCOVA (analysis of covariance) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches Table of Contents Lord’s paradox 1 Lord, 1967 understanding the two approaches comparison of the approaches 2 psychometricians on change scores attempts using causal framework a closer look ADM study 3 Data Modeling curriculum analysis of the data Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches the two approaches change score approach: intuitive, easy to interpret John Snow‘s finding that the cholera was a water-borne infectious disease regressor variable approach: as long as temporal order allows, seems reasonable to learn from the data extreme example: pretest and posttest of weight group 1: mice group 2: elephants Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches the two approaches change score approach: intuitive, easy to interpret John Snow‘s finding that the cholera was a water-borne infectious disease regressor variable approach: as long as temporal order allows, seems reasonable to learn from the data extreme example: pretest and posttest of weight group 1: mice group 2: elephants Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches the two approaches change score approach: intuitive, easy to interpret John Snow‘s finding that the cholera was a water-borne infectious disease regressor variable approach: as long as temporal order allows, seems reasonable to learn from the data extreme example: pretest and posttest of weight group 1: mice group 2: elephants Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches the two approaches change score approach: intuitive, easy to interpret John Snow‘s finding that the cholera was a water-borne infectious disease regressor variable approach: as long as temporal order allows, seems reasonable to learn from the data extreme example: pretest and posttest of weight group 1: mice group 2: elephants Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches Lord’s paradox Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches treatment assigned at random Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches on Lord, 1967 Lord (1967): “... there are as many different explanations as there are explainers” “... there simply is no logical or statistical procedure that can be counted on to make proper allowances for uncontrolled preexisting differences between groups” Senn (2008): “In a disturbing paper in the Psychological Bulletin in 1967, Lord considered a case..,” Rubin, Stuart & Zanutto (2003): “A Classic Example of Poorly Formulated Causal Assessment—Lord’s paradox” Wainer & Brown (2007): “..by far, the most difficult paradox to disentangle and requires clear thinking” change score method: underdog in educational and social methodology (but not in econometrics) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches on Lord, 1967 Lord (1967): “... there are as many different explanations as there are explainers” “... there simply is no logical or statistical procedure that can be counted on to make proper allowances for uncontrolled preexisting differences between groups” Senn (2008): “In a disturbing paper in the Psychological Bulletin in 1967, Lord considered a case..,” Rubin, Stuart & Zanutto (2003): “A Classic Example of Poorly Formulated Causal Assessment—Lord’s paradox” Wainer & Brown (2007): “..by far, the most difficult paradox to disentangle and requires clear thinking” change score method: underdog in educational and social methodology (but not in econometrics) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches on Lord, 1967 Lord (1967): “... there are as many different explanations as there are explainers” “... there simply is no logical or statistical procedure that can be counted on to make proper allowances for uncontrolled preexisting differences between groups” Senn (2008): “In a disturbing paper in the Psychological Bulletin in 1967, Lord considered a case..,” Rubin, Stuart & Zanutto (2003): “A Classic Example of Poorly Formulated Causal Assessment—Lord’s paradox” Wainer & Brown (2007): “..by far, the most difficult paradox to disentangle and requires clear thinking” change score method: underdog in educational and social methodology (but not in econometrics) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches on Lord, 1967 Lord (1967): “... there are as many different explanations as there are explainers” “... there simply is no logical or statistical procedure that can be counted on to make proper allowances for uncontrolled preexisting differences between groups” Senn (2008): “In a disturbing paper in the Psychological Bulletin in 1967, Lord considered a case..,” Rubin, Stuart & Zanutto (2003): “A Classic Example of Poorly Formulated Causal Assessment—Lord’s paradox” Wainer & Brown (2007): “..by far, the most difficult paradox to disentangle and requires clear thinking” change score method: underdog in educational and social methodology (but not in econometrics) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study understanding the two approaches on Lord, 1967 Lord (1967): “... there are as many different explanations as there are explainers” “... there simply is no logical or statistical procedure that can be counted on to make proper allowances for uncontrolled preexisting differences between groups” Senn (2008): “In a disturbing paper in the Psychological Bulletin in 1967, Lord considered a case..,” Rubin, Stuart & Zanutto (2003): “A Classic Example of Poorly Formulated Causal Assessment—Lord’s paradox” Wainer & Brown (2007): “..by far, the most difficult paradox to disentangle and requires clear thinking” change score method: underdog in educational and social methodology (but not in econometrics) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study three camps debates over this paradox spread into mainly three directions: psychometricians: debates over reliability, measurement error, regression to the mean (Cronbach & Furby, 1970; Linn & Slinde, 1977 vs. Rogosa & Willett, 1983; Zimmerman & Williams, 1982) whose causal framework is better armed to explain the paradox (Holland & Rubin, 1982; Wainer & Brown, 2007; Pearl, 2014) ”it depends” camp (Kenny, 1975, Allison, 1990) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study three camps debates over this paradox spread into mainly three directions: psychometricians: debates over reliability, measurement error, regression to the mean (Cronbach & Furby, 1970; Linn & Slinde, 1977 vs. Rogosa & Willett, 1983; Zimmerman & Williams, 1982) whose causal framework is better armed to explain the paradox (Holland & Rubin, 1982; Wainer & Brown, 2007; Pearl, 2014) ”it depends” camp (Kenny, 1975, Allison, 1990) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study three camps debates over this paradox spread into mainly three directions: psychometricians: debates over reliability, measurement error, regression to the mean (Cronbach & Furby, 1970; Linn & Slinde, 1977 vs. Rogosa & Willett, 1983; Zimmerman & Williams, 1982) whose causal framework is better armed to explain the paradox (Holland & Rubin, 1982; Wainer & Brown, 2007; Pearl, 2014) ”it depends” camp (Kenny, 1975, Allison, 1990) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study three camps debates over this paradox spread into mainly three directions: psychometricians: debates over reliability, measurement error, regression to the mean (Cronbach & Furby, 1970; Linn & Slinde, 1977 vs. Rogosa & Willett, 1983; Zimmerman & Williams, 1982) whose causal framework is better armed to explain the paradox (Holland & Rubin, 1982; Wainer & Brown, 2007; Pearl, 2014) ”it depends” camp (Kenny, 1975, Allison, 1990) Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores Table of Contents Lord’s paradox 1 Lord, 1967 understanding the two approaches comparison of the approaches 2 psychometricians on change scores attempts using causal framework a closer look ADM study 3 Data Modeling curriculum analysis of the data Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Cronbach & Furby (1970): “It appears that investigators who ask questions regarding gain scores would ordinarily be better advised to frame their questions in other ways.” Linn & Slinde (1977): “Problems in measuring change abound and the virtues in doing so are hard to find.” main “issues”: gain scores will be higher for individuals with a lower pretest (“unfairness”) unreliable: Gulliksen’s (1950) formula Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Cronbach & Furby (1970): “It appears that investigators who ask questions regarding gain scores would ordinarily be better advised to frame their questions in other ways.” Linn & Slinde (1977): “Problems in measuring change abound and the virtues in doing so are hard to find.” main “issues”: gain scores will be higher for individuals with a lower pretest (“unfairness”) unreliable: Gulliksen’s (1950) formula Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Cronbach & Furby (1970): “It appears that investigators who ask questions regarding gain scores would ordinarily be better advised to frame their questions in other ways.” Linn & Slinde (1977): “Problems in measuring change abound and the virtues in doing so are hard to find.” main “issues”: gain scores will be higher for individuals with a lower pretest (“unfairness”) unreliable: Gulliksen’s (1950) formula Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Cronbach & Furby (1970): “It appears that investigators who ask questions regarding gain scores would ordinarily be better advised to frame their questions in other ways.” Linn & Slinde (1977): “Problems in measuring change abound and the virtues in doing so are hard to find.” main “issues”: gain scores will be higher for individuals with a lower pretest (“unfairness”) unreliable: Gulliksen’s (1950) formula Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Cronbach & Furby (1970): “It appears that investigators who ask questions regarding gain scores would ordinarily be better advised to frame their questions in other ways.” Linn & Slinde (1977): “Problems in measuring change abound and the virtues in doing so are hard to find.” main “issues”: gain scores will be higher for individuals with a lower pretest (“unfairness”) unreliable: Gulliksen’s (1950) formula Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Willett (1997): “change and previous status will always be related since current status is the product of prior change” Rogosa, Brandt, & Zimowski (1982): “The correlation between true change and true initial status (zero or otherwise) is an interesting fact of life.” (..but no more than that) Allison (1990): “regression toward the mean is not an issue when the objective is to compare two groups” [assuming equivalence at baseline] counterexample in education: those who are high in the initial status might be better suited to understand the new instruction and gain more than those with a lower initial status Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Willett (1997): “change and previous status will always be related since current status is the product of prior change” Rogosa, Brandt, & Zimowski (1982): “The correlation between true change and true initial status (zero or otherwise) is an interesting fact of life.” (..but no more than that) Allison (1990): “regression toward the mean is not an issue when the objective is to compare two groups” [assuming equivalence at baseline] counterexample in education: those who are high in the initial status might be better suited to understand the new instruction and gain more than those with a lower initial status Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Willett (1997): “change and previous status will always be related since current status is the product of prior change” Rogosa, Brandt, & Zimowski (1982): “The correlation between true change and true initial status (zero or otherwise) is an interesting fact of life.” (..but no more than that) Allison (1990): “regression toward the mean is not an issue when the objective is to compare two groups” [assuming equivalence at baseline] counterexample in education: those who are high in the initial status might be better suited to understand the new instruction and gain more than those with a lower initial status Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study psychometricians on change scores psychometricians on change scores Willett (1997): “change and previous status will always be related since current status is the product of prior change” Rogosa, Brandt, & Zimowski (1982): “The correlation between true change and true initial status (zero or otherwise) is an interesting fact of life.” (..but no more than that) Allison (1990): “regression toward the mean is not an issue when the objective is to compare two groups” [assuming equivalence at baseline] counterexample in education: those who are high in the initial status might be better suited to understand the new instruction and gain more than those with a lower initial status Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Table of Contents Lord’s paradox 1 Lord, 1967 understanding the two approaches comparison of the approaches 2 psychometricians on change scores attempts using causal framework a closer look ADM study 3 Data Modeling curriculum analysis of the data Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Neyman-Rubin framework Rubin, et al., (2003); Holland & Rubin (1983): Lord’s example was a “poorly formulated causal assessment” since the potential outcome under the control diet is missing “...researcher investigating gain wouldn’t know if changes in scores would have occurred with no treatment anyway” the hypothetical researcher in Lord (1967) is interested in gender differences and not in the effect of the diet Lord (1967): “differential effect” of the diet under the Neyman-Rubin (a.k.a. potential outcomes) causal framework the effect of gender cannot be a causal research question Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Neyman-Rubin framework Rubin, et al., (2003); Holland & Rubin (1983): Lord’s example was a “poorly formulated causal assessment” since the potential outcome under the control diet is missing “...researcher investigating gain wouldn’t know if changes in scores would have occurred with no treatment anyway” the hypothetical researcher in Lord (1967) is interested in gender differences and not in the effect of the diet Lord (1967): “differential effect” of the diet under the Neyman-Rubin (a.k.a. potential outcomes) causal framework the effect of gender cannot be a causal research question Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Neyman-Rubin framework Rubin, et al., (2003); Holland & Rubin (1983): Lord’s example was a “poorly formulated causal assessment” since the potential outcome under the control diet is missing “...researcher investigating gain wouldn’t know if changes in scores would have occurred with no treatment anyway” the hypothetical researcher in Lord (1967) is interested in gender differences and not in the effect of the diet Lord (1967): “differential effect” of the diet under the Neyman-Rubin (a.k.a. potential outcomes) causal framework the effect of gender cannot be a causal research question Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Neyman-Rubin framework Rubin, et al., (2003); Holland & Rubin (1983): Lord’s example was a “poorly formulated causal assessment” since the potential outcome under the control diet is missing “...researcher investigating gain wouldn’t know if changes in scores would have occurred with no treatment anyway” the hypothetical researcher in Lord (1967) is interested in gender differences and not in the effect of the diet Lord (1967): “differential effect” of the diet under the Neyman-Rubin (a.k.a. potential outcomes) causal framework the effect of gender cannot be a causal research question Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Neyman-Rubin framework Rubin, et al., (2003); Holland & Rubin (1983): Lord’s example was a “poorly formulated causal assessment” since the potential outcome under the control diet is missing “...researcher investigating gain wouldn’t know if changes in scores would have occurred with no treatment anyway” the hypothetical researcher in Lord (1967) is interested in gender differences and not in the effect of the diet Lord (1967): “differential effect” of the diet under the Neyman-Rubin (a.k.a. potential outcomes) causal framework the effect of gender cannot be a causal research question Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study attempts using causal framework Neyman-Rubin framework Rubin, et al., (2003); Holland & Rubin (1983): Lord’s example was a “poorly formulated causal assessment” since the potential outcome under the control diet is missing “...researcher investigating gain wouldn’t know if changes in scores would have occurred with no treatment anyway” the hypothetical researcher in Lord (1967) is interested in gender differences and not in the effect of the diet Lord (1967): “differential effect” of the diet under the Neyman-Rubin (a.k.a. potential outcomes) causal framework the effect of gender cannot be a causal research question Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look Table of Contents Lord’s paradox 1 Lord, 1967 understanding the two approaches comparison of the approaches 2 psychometricians on change scores attempts using causal framework a closer look ADM study 3 Data Modeling curriculum analysis of the data Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look regressor variable model Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j OLS yields unbiased estimates assuming: ǫ j uncorrelated with W j and Y 1 j correct specification i.i.d. no measurement error Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look change score model (following Allison, 1990) assume G j is binary: 1 if person j ends up in the treatment group; 0 otherwise pre: Y 1 j = β 0 + δ G j + ǫ 1 j δ : group differences that are stable post: Y 2 j = β 0 + β 1 + δ G j + β 2 W j + ǫ 2 j W j is treatment indicator G j = W j (collinear) β 1 represents the change that is occurring in both groups (e.g., gained knowledge during a school-year) Y 2 j − Y 1 j = ( β 0 − β 0 ) + β 1 + ( δ G j − δ G j ) + β 2 W j + ( ǫ 2 j − ǫ 1 j ) ∆ Y j = β 1 + β 2 W j + ǫ ∆ j assuming ǫ ∆ is not correlated with W j , OLS is consistent and j hence the estimates are unbiased Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? change score method: Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j rewrite: Y 2 j = β 1 + β 2 W j + (1) Y 1 j + ǫ ∆ j some say/imply that this is a special case of the: Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ j see for instance: Hedeker & Gibbons, 2006 (p. 8) Van Breukelen, 2013, (p. 903) Gelman & Hill, 2006 (p. 177) “an unnecessary assumption, namely, that [ β 3 = 1] ” Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
Lord’s paradox comparison of the approaches ADM study a closer look special case? Y 2 j = β 1 + β 2 W j + β 3 Y 1 j + ǫ ∆ j inconsistent estimates since ǫ ∆ is negatively correlated with j Y 1 j by construction Y 2 j − Y 1 j = β 1 + β 2 W j + ǫ ∆ j not a special case the two approaches represent two completely different models! overlooking this crucial distinction has been the most common error in comparisons of the two approaches any discussion of Lord’s paradox that does not acknowledge this distinction is likely to be misleading Perman Gochyyev GSE, UC Berkeley BEAR Center Evaluating the Treatment Effect in the ADM Study and Lord’s Paradox
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