The statistical analysis for a 2-arm randomised controlled trial (RCT) with a baseline outcome followed by a few assessments at fixed follow-up times typically invokes traditional analytic methods (eg, analysis of covariance (ANCOVA), longitudinal data analysis (LDA)). ‘Constrained’ longitudinal data analysis (cLDA) is a well-established unconditional technique that constrains means of baseline to be equal between arms. We use an analysis of fasting lipid profiles from the Group Medical Clinics (GMC) longitudinal RCT on patients with diabetes to illustrate applications of ANCOVA, LDA and cLDA to demonstrate theoretical concepts of these methods including the impact of missing data.

For the analysis of the illustrated example, all models were fit using linear mixed models to participants with only complete data and to participants using all available data.

With complete data (n=195), 95% CI coverage are equivalent for ANCOVA and cLDA with an estimated 11.2 mg/dL (95% CI −19.2 to −3.3; p=0.006) lower mean low-density lipoprotein (LDL) cholesterol in GMC compared with usual care. With all available data (n=233), applying the cLDA model yielded an LDL improvement of 8.9 mg/dL (95% CI −16.7 to −1.0; p=0.03) for GMC compared with usual care. The less efficient, LDA analysis yielded an LDL improvement of 7.2 mg/dL (95% CI −17.2 to 2.8; p=0.15) for GMC compared with usual care.

Under reasonable missing data assumptions, cLDA will yield efficient treatment effect estimates and robust inferential statistics. It may be regarded as the method of choice over ANCOVA and LDA.

Clarification of the statistical methods available for longitudinal randomised controlled trials (RCTs) as well as analysis recommendations is warranted.

In many longitudinal RCTs, participants are measured at baseline, then at the same follow-up occasions with a small number of follow-ups, for example, 2–4. In this design, how should baseline values be handled?

In practical applications, constrained longitudinal data analysis, an appropriate generalisation of analysis of covariance, is the most straightforward to implement and under reasonable missing data assumptions will yield robust estimates of treatment effect differences and valid inferential statistics.

In a recent longitudinal randomised controlled trial (RCT) designed to examine the effect of Group Medical Clinics (GMC) on cardiovascular outcomes in patients with diabetes, the statistical inference on the effect of the GMC intervention on low-density lipoprotein (LDL) levels was dependent on the method of analysis applied.

In many longitudinal RCTs, research participants are measured at the same follow-up measurement occasions and the number of follow-up occasions is small, for example, 2–4. Typically, baseline outcomes are measured prior to randomisation. In analyses, baseline outcomes can be ignored, used to calculate change scores, used conditionally as covariates or can be part of the outcome response vector. In this design, how should baseline values be handled? Is the question of interest conditional, given one's baseline what is the difference in outcome between treatment arms, or is the question unconditional, what is the treatment difference in change? Or does it matter? These fundamental questions and their implications on the method of analysis were introduced over 50 years ago in a sentinel paper by Lord

As the issue is addressed here, a major complication requiring consideration in the analysis strategy of a longitudinal RCT is missing data. Attrition (dropout) or intermittent missingness occurs even in studies with few measurement occasions as we are addressing. Reducing the quantity of missing data in an RCT enhances the reliability of results;

The general consensus in the statistical literature for a longitudinal RCT is that the conditional approach, that is, analysis of covariance (ANCOVA), is the most powerful and robust method

In a longitudinal RCT of the type addressed here, the outcome of interest is measured at each of the defined assessment periods of the trial (each participant in each arm of the trial). In this setting define the outcome as

In the more general longitudinal RCT study design with multiple but few follow-up time points, response profile modelling

A fundamental difference between a conditional or unconditional analysis is in the modelling of the pre-treatment assessment. In unconditional analysis, baseline is part of the response vector requiring additional assumptions for modelling baseline. In an LDA, there are no modelling constraints on the baseline; separate baseline means are assumed and fit for each randomised group. The general test used for treatment difference over time in LDA is equivalent to a change score type of analysis; comparing change from baseline to follow-up between randomised groups. In contrast for cLDA baseline means are constrained to be equal between the randomised groups; a common baseline mean is assumed and fit across randomised groups. In an RCT, baseline precedes any treatment deliverance and under expectation the baseline means are equal. The test for treatment difference over time in the cLDA is essentially equivalent to a test for treatment difference in an ANCOVA when no outcome data are missing.

An ANCOVA model in a two-arm (j=1, 2) RCT with a pre/post design will have study outcome measures at baseline and one follow-up time (t=0,1). However, there will only be one response variable per participant (i=1,…,n_{j}) as the baseline (Y_{0}) is a covariate in the model. The model is written as:

The marginal mean at follow-up time (t=1) is conditional on the baseline

An LDA model in a two-arm (j=1, 2) RCT with a pre/post design will have study outcome measures at both the baseline and follow-up time (t=0,1). The model is written as:

The parameter γ is the difference in baseline means at time (t=0) between arms, τ is the mean difference in change from baseline to follow-up for arm (j=1), and δ is the difference in change from baseline to follow-up between arms.

A cLDA model in a two-arm (j=1, 2) RCT with a pre/post design will have study outcome measures at both the baseline and follow-up time (t=0,1). The model is written as:

The parameter

Similarly, for reasonably sized trials the variance of the treatment effect differences will be approximately equivalent between ANCOVA and cLDA models (see online

Frison and Pocock

Comparison of variance of treatment difference estimates over the range of correlations between pre and post measurements for LDA, cLDA, ANCOVA and SPO methods; plot generated from variance estimate formulas given in online

The conditional and unconditional models described above are easily extendible from the pre/post type design with only one follow-up time point to multiple follow-up time points by additional dummy variables to represent follow-up times. Similarly, comparison of models and estimates from the pre/post design apply to the follow-up time points. In the longitudinal RCTs with few measurement occasions, the comparison of interest is generally the treatment difference at the last follow-up time point (T)—the theory as described above for the post-time measurement in a pre/post design applies to analysing the differences between treatment arms at any specific follow-up time.

We present an analysis of fasting lipid profiles from the GMC longitudinal RCT on patients with diabetes to illustrate applications of ANCOVA, LDA and cLDA models and demonstrate theoretical concepts described above including the impact of missing data.

For illustration, we focus analysis on the baseline and 12-month LDL cholesterol (LDL-C) measurements. For notation, the two arms will be denoted with j=G for GMC or j=U for usual care and time with t=0 for baseline and t=12 for 12 months. All analyses were conducted using SAS V.9.2 (SAS Institute, Cary, North Carolina, USA).

For completeness of analyses approaches, in addition to the ANCOVA, LDA and cLDA models, we also conducted simple post only (SPO) and simple change score analysis (SACS). For SPO, we compared mean 12-month LDL-C (

The first set of analyses was applied to participants that had both baseline and 12-month measurements (completers) to demonstrate theoretical comparisons of models as described without the added complication of missing data.

Among the 239 patients randomised in the study, 195 participants had LDL-C measurements at both baseline and 12 months. All methods yield statistically significant differences in LDL-C between arms (

Completers only (n=195 participants)—pre/post analyses

Model | Outcome (Y_{t} and/or C_{t}) | GMC (n=117) | Usual care (n=78) | GMC vs usual care (95% CI) | p Value |
---|---|---|---|---|---|

Post-only | 12 months (Y_{12}) | 81.9 | 94.1 | −12.1 (−21.5 to −2.7) | 0.01 |

SACS | 12 months_{12}) | −12.9 | −2.8 | −10.1 (−20.2 to 0.0) | 0.05 |

ANCOVA | 12 months (Y_{12}) | 82.3 | 93.5 | −11.2 (−19.2 to −3.3) | 0.006 |

LDA | Baseline (Y_{0}) | 94.8 | 96.9 | ||

12 months (Y_{12}) | 81.9 | 94.1 | −10.1 (−20.2 to 0.0) | 0.05 | |

cLDA | Baseline (Y_{0}) | 95.7 | 95.7 | ||

12 months (Y_{12}) | 82.3 | 93.5 | −11.2 (−19.2 to −3.3) | 0.006 |

ANCOVA, analysis of covariance; cLDA, constrained longitudinal data analysis; GMC, Group Medical Clinics; LDA, longitudinal data analysis; SACS, simple change score analysis.

The second set of analyses was applied to all participants (ie, including those with either missing baseline or 12-month measurements) to compare methods and illustrate the impact of missing data. SPO participants with missing 12-month LDL-C are deleted and SACS participants with either missing baseline or end of study measurements are deleted. Similarly, for ANCOVA with only two time points, participants with missing baseline or 12-month measurements are deleted. For LDA and cLDA, all available data were used; no participants were deleted due to missing data. The estimation procedure used in the mixed model framework for longitudinal analysis yields unbiased estimates of parameters when missing outcomes are assumed to be ignorable, that is, when missing values are related to either observed covariates or response variables but not to unobserved variables.

Among the 239 patients randomised, 6 participants had no baseline or 12-month LDL-C, so they are excluded yielding 233 patients for the all participants analysis. The estimated treatment differences diverge across analysis methods as well as statistical significance using a α=0.05 (

All available data (n=233 participants)—pre/post analyses

Model | Outcome (Y_{t} and/or C_{t}) | N | GMC | Usual care | GMC vs usual care (95% CI) | p Value |
---|---|---|---|---|---|---|

Post-only | 12 months (Y_{12}) | 204 | 89.7 | 96.7 | −6.9 (−14.2 to 0.4) | 0.07 |

SACS | 12 months_{12}) | 195 | −12.9 | −2.8 | −10.1 (−22.0 to −0.8) | 0.05 |

ANCOVA | 12 months (Y_{12}) | 195 | 83.4 | 94.6 | −11.2 (−19.2 to −3.3) | 0.006 |

LDA* | Baseline (Y_{0}) | 233 | 96.7 | 99.6 | ||

12 months (Y_{12}) | 83.5 | 93.6 | −7.2 (−17.2 to 2.8) | 0.15 | ||

cLDA* | Baseline (Y_{0}) | 233 | 98.0 | 98.0 | ||

12 months (Y_{12}) | 84.0 | 92.9 | −8.9 (−16.7 to −1.0) | 0.03 |

*Baseline LDL-C is missing for 9 participants and 12-month LDL-C is missing for 29 participants.

ANCOVA, analysis of covariance; cLDA, constrained longitudinal data analysis; GMC, Group Medical Clinics; LDA, longitudinal data analysis; LDL-C, low-density lipoprotein cholesterol; SACS, simple change score analysis.

As shown for ANCOVA and cLDA, LDA and SACS methods estimated differences are discrepant due to missing data and assumptions that are made about missing data. LDA is using all available data and fit using mixed-effects model, whereas in SACS participants with missing data are deleted. Similar to completers analysis, LDA methods yield wider 95% CI than both cLDA and ANCOVA.

The illustrative example presented above clearly demonstrates the statistical theory described for the analysis methods and the impact of missing data

The choice of whether to use multiple imputation is up to the analyst and based on the untestable assumption that all predictors of missing data can be included in the mixed-effect model. In many longitudinal RCTs it is a reasonable assumption that treatment group assignment, time of assessment and available outcome assessments along with potential baseline stratification variables would be the predictors of missing outcome data. If the number of potential predictor variables of missing data is expansive and unreasonable to include in the mixed-effects model then it may be necessary to perform a multiple imputation analysis.

Multiple imputation is a more complicated process for handling missing data that is also based on the untestable assumption that all predictors of missing data are included in the imputation model.

For ease of implementation and robustness of results

It does not appear widely appreciated that cLDA can often be regarded as the method of choice for the analysis of a longitudinal RCT with few measurement occasions. Except for small samples it is equivalent to ANCOVA when there are no missing data, and cLDA is an appropriate generalisation of ANCOVA under reasonable missing data assumptions. Without question, potential baseline imbalance between treatment arms has implications for the analysis of longitudinal RCTs and is a source of confusion. In an RCT, baseline differences can be attributed to random chance assuming there were no problems or issues with the randomisation process and no significant measurement error issues. LDA or CHANGE score analysis is sometimes viewed as a more intuitive analysis; however, for an RCT most often the best method is an ANCOVA as far as bias, precision and power.

cLDA is sometimes erroneously viewed as more problematic when there is baseline imbalance in outcomes between treatment arms. However, cLDA and ANCOVA are equivalent when analysing complete data. cLDA generalises the ANCOVA approach and both are superior to an LDA in many cases. Therefore, the primary analytic issue is not necessarily whether or not to perform conditional analysis. ANCOVA is a conditional analysis and cLDA is an unconditional analysis, yet both are powerful methods that can be applied to examine treatment differences over time in a longitudinal RCT. In practical applications, cLDA is the most straightforward to implement and under reasonable missing data assumptions will yield robust estimates of treatment effect differences and inferential statistics. In most cases, it is the method of choice.

The primary author is a senior statistician in the Biostatistics Unit of the Durham VA Health Services Research and Development group and an Associate Professor in the Department of Biostatistics and Bioinformatics at Duke University Medical Center. All listed authors have contributed to the design and preparation of the manuscript (CJC, DE and RFW). All data analyses were conducted by CJC.

The study was funded by the US Department of Veterans Affairs Health Services Research and Development Service IIR 03-084.

None declared.

Not commissioned; externally peer reviewed.

No additional data are available.