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Using ANCOVA when groups differ on the covariate is controversial, although Tabachnick and Fidell write that this is a plausible function of ANCOVA in quasi-experimental (or observational) studies. As they state:

The second use of ANCOVA commonly occurs in nonexperimental situations when subjects cannot be randomly assigned to treatments. ANCOVA is used as a statistical matching procedure, although interpretation is fraught with difficulty [...]. ANCOVA is used primarily to adjust goup means to what they would be if all subjects scored identically on the CV(s). Differences between subjects on CVs are removed so that, presumably, the only differences that remain are related to the effects of the grouping IV(s). (Differences could also, of course, be due to attributes that have not been used as CVs.) This second application of ANCOVA is primarily for descriptive model building: the CV enhances prediction of the DV, but there is no implication of causality. If the research question to be answered involves causality, ANCOVA is no substitute for running an experiment.

Moreover, in this question the same issue was addressed, and the use of ANCOVA for intact groups was encouraged.

My question is: in these situations, in which the assumption of independence of the covariate from the treatment variable is violated, what are the assumptions? For example, must the covariate be correlated with the dependent variable inside the groups? Or are the assumptions simply the same as for ANOVA?

Using ANCOVA when groups differ on the covariate is controversial, although Tabachnick and Fidell write that this is a plausible function of ANCOVA in quasi-experimental (or observational) studies. As they state:

The second use of ANCOVA commonly occurs in nonexperimental situations when subjects cannot be randomly assigned to treatments. ANCOVA is used as a statistical matching procedure, although interpretation is fraught with difficulty [...]. ANCOVA is used primarily to adjust goup means to what they would be if all subjects scored identically on the CV(s). Differences between subjects on CVs are removed so that, presumably, the only differences that remain are related to the effects of the grouping IV(s). (Differences could also, of course, be due to attributes that have not been used as CVs.) This second application of ANCOVA is primarily for descriptive model building: the CV enhances prediction of the DV, but there is no implication of causality. If the research question to be answered involves causality, ANCOVA is no substitute for running an experiment.

Moreover, in this question the same issue was addressed, and the use of ANCOVA for intact groups was encouraged.

My question is: in these situations, in which the assumption of independence of the covariate from the treatment variable is violated, what are the assumptions? For example, must the covariate be correlated with the dependent variable inside the groups? Or are the assumptions simply the same as for ANOVA?

Using ANCOVA when groups differ on the covariate is controversial, although Tabachnick and Fidell write that this is a plausible function of ANCOVA in quasi-experimental (or observational) studies. As they state:

The second use of ANCOVA commonly occurs in nonexperimental situations when subjects cannot be randomly assigned to treatments. ANCOVA is used as a statistical matching procedure, although interpretation is fraught with difficulty [...]. ANCOVA is used primarily to adjust goup means to what they would be if all subjects scored identically on the CV(s). Differences between subjects on CVs are removed so that, presumably, the only differences that remain are related to the effects of the grouping IV(s). (Differences could also, of course, be due to attributes that have not been used as CVs.) This second application of ANCOVA is primarily for descriptive model building: the CV enhances prediction of the DV, but there is no implication of causality. If the research question to be answered involves causality, ANCOVA is no substitute for running an experiment.

Moreover, in this question the same issue was addressed, and the use of ANCOVA for intact groups was encouraged.

My question is: in these situations, in which the assumption of independence of the covariate from the treatment variable is violated, what are the assumptions? For example, must the covariate be correlated with the dependent variable inside the groups?

Using ANCOVA when groups differ on the covariate is controversial, although Tabachnick and Fidell write that this is a plausible function of ANCOVA in quasi-experimental (or observational) studies. As they state:

The second use of ANCOVA commonly occurs in nonexperimental situations when subjects cannot be randomly assigned to treatments. ANCOVA is used as a statistical matching procedure, although interpretation is fraught with difficulty [...]. ANCOVA is used primarily to adjust goup means to what they would be if all subjects scored identically on the CV(s). Differences between subjects on CVs are removed so that, presumably, the only differences that remain are related to the effects of the grouping IV(s). (Differences could also, of course, be due to attributes that have not been used as CVs.) This second application of ANCOVA is primarily for descriptive model building: the CV enhances prediction of the DV, but there is no implication of causality. If the research question to be answered involves causality, ANCOVA is no substitute for running an experiment.

Moreover, in this question the same issue was addressed, and the use of ANCOVA for intact groups was encouraged.

My question is: in these situations, in which the assumption of independence of the covariate from the treatment variable is violated, what are the assumptions? For example, must the covariate be correlated with the dependent variable inside the groups? Or are the assumptions simply the same as for ANOVA?

Using ANCOVA when groups differ on the covariate is controversial, although Tabachnick and Fidell write that this is a plausible function of ANCOVA in quasi-experimental (or observational) studies. As they state:

The second use of ANCOVA commonly occurs in nonexperimental situations when subjects cannot be randomly assigned to treatments. ANCOVA is used as a statistical matching procedure, although interpretation is fraught with difficulty [...]. ANCOVA is used primarily to adjust goup means to what they would be if all subjects scored identically on the CV(s). Differences between subjects on CVs are removed so that, presumably, the only differences that remain are related to the effects of the grouping IV(s). (Differences could also, of course, be due to attributes that have not been used as CVs.) This second application of ANCOVA is primarily for descriptive model building: the CV enhances prediction of the DV, but there is no implication of causality. If the research question to be answered involves causality, ANCOVA is no substitute for running an experiment.

Moreover, in this question question the same issue was addressed.

My question is: in these situations, in which the assumption of independence of the covariate from the treatment variable is violated, what are the assumptions? For example, must the covariate be correlated with the dependent variable inside the groups?

Using ANCOVA when groups differ on the covariate is controversial, although Tabachnick and Fidell write that this is a plausible function of ANCOVA in quasi-experimental (or observational) studies. As they state:

The second use of ANCOVA commonly occurs in nonexperimental situations when subjects cannot be randomly assigned to treatments. ANCOVA is used as a statistical matching procedure, although interpretation is fraught with difficulty [...]. ANCOVA is used primarily to adjust goup means to what they would be if all subjects scored identically on the CV(s). Differences between subjects on CVs are removed so that, presumably, the only differences that remain are related to the effects of the grouping IV(s). (Differences could also, of course, be due to attributes that have not been used as CVs.) This second application of ANCOVA is primarily for descriptive model building: the CV enhances prediction of the DV, but there is no implication of causality. If the research question to be answered involves causality, ANCOVA is no substitute for running an experiment.

Moreover, in this question the same issue was addressed, and the use of ANCOVA for intact groups was encouraged.

My question is: in these situations, in which the assumption of independence of the covariate from the treatment variable is violated, what are the assumptions? For example, must the covariate be correlated with the dependent variable inside the groups?