4 Variables (1) Age (2) Sex (3) Mental status (MMSE) (4) Years of education (5) Economic suﬃciency (6) ADLs (7) Physical health (8) Mental heath (9) Self-eﬃcacy (10) Social resources (11) Social provisions Table 1: Descriptives and ANOVA results for study variables. Private homes M (SD) 99.76 (1.71) 76.2% female 20.43 (7.09) 11.65 (3.87) 2.56 (.63) 37.11 (7.43) 1.87 (.76) 2.02 (.62) 5.66 (2.09) 1.81 (.78) 35.19 (2.15) Assisted living facilities M (SD) 100 (2.00) 86% female 19.44 (7.87) 13 (3.05) 2.67 (.48) 37.42 (7.03) 1.85 (.62) 1.94 (.57) 5.64 (2.63) 1.83 (.75) 35.24 (2.55) Nursing homes M (SD) 101 (2.10) 87.6% female 11.37 (8.24) 11.15 (3.59) 2.73 (.47) 33.98 (8.88) 1.82 (.91) 1.73 (.76) 5.50 (1.98) 1.96 (.84) 34.39 (2.73) Journal of Aging Research One-way ANOVA ∗ ∗ ∗ ∗∗ ∗ ∗ Private homes diﬀered from nursing homes, P < .05; ∗∗ Private homes and assisted living facilities diﬀered from nursing homes, P < .05; ∗∗∗ Each was signiﬁcantly diﬀerent from the other, P < .05. and social provisions between octogenarians and cente- narians living in private homes and among centenarians residing in private homes, assisted living facilities, and nursing homes, we conducted our analyses by subgroup. Based upon Antonucci and colleagues’ model [13], we tested a path analytic model for each subgroup. We used path analysis for a number of reasons. First, because of our limited sample size, we employed manifest rather than latent variables to reduce the number of parameters estimated in our models. Second, if necessary, we also wanted to test for model equivalency across our subgroups. Third, our hypotheses called for tests of directionality with self-eﬃcacy as a mediating mechanism. Working with very old adults presents numerous challenges, one of which is locating and assessing participants who often present with numerous hearing, visual, and other medical conditions, in addition to mortality concerns over time [7], resulting in smaller sample sizes. The literature on sample size and covariance structure modeling has addressed the concern of sample size and the number of parameters estimated [52–54]. Jackson concluded that when small sample sizes (he compared sizes of 50, 100, 200, 400, and 800) are encountered, assessments of model ﬁt such as chi-square goodness-of-ﬁt and RMSEA, for example, are most sensitive to misspeciﬁcation and therefore, recommended. In addition, Herzog and Boomsma conducted a Monte Carlo study demonstrating that Swain- corrected estimators were robust to small sample sizes and recommended their use for small sample size research [52]. We used their syntax for use with R software for each of our ﬁnal models presented in the Section 3 below, and no diﬀerences in ﬁt were found between what we reported and the Swain-corrected estimations. Thus, a path model allowed us to examine model ﬁt, model equivalency between groups, and indirect eﬀects through the mediating mechanism of self-eﬃcacy. The literature on multiple-sample testing in structural equation modeling argues for establishing a baseline or best- ﬁtting model for each sample ﬁrst [55]. Kline extends the point and argues that unless an unconstrained model ﬁts well across samples, it makes little sense to test additional constraints [56, page 295]. We ﬁrst speciﬁed and tested the hypothesized support-eﬃcacy model, determining the best model ﬁt for each subgroup (a series of nested model tests examining change in chi-square for each nested model). Analyses were conducted with Mplus Version 5.0 [57] using full-information maximum likelihood (FIML) to handle missing data; overall model ﬁt was assessed by employing the Satorra-Bentler chi-square test statistic that is robust to nonnormality of measures, referred to as the MLR χ2 in Mplus. Model evaluation was based on the chi-square goodness of ﬁt test and other ﬁt indices: the comparative ﬁt index—CFI [58], root mean squared error of approximation (RMSEA, [59]), and the standardized root mean squared residual (SRMR). Values close to .95 for CFI, .08 for SRMR, and .06 for RMSEA suggest that good ﬁt between the observed data and the hypothesized model exists [60, 61]. Nested model testing was conducted according to the procedure outlined in Mplus [57]. Control variables included sex, mental status, perceived economic suﬃciency, and activities of daily living. 3. Results 3.1. Centenarians in Private Homes. A base model with no degrees of freedom (all possible paths were estimated) was tested. The regression of mental health on social resources was not signiﬁcant, (β = .13; P > .05), neither was the regression of mental health on self-eﬃcacy (β = .05; P > .05). These paths were deleted, and the nested model ﬁt the data well: MLR χ2 (2, N = 126) = 2.36, P = .31, CFI = .99; RMSEA = .04; SRMR = .02. In this model, all speciﬁed paths were signiﬁcant (one-tailed tests); however physical health regressed on social provisions (β = .20; P > .05) was the weakest path. Thus, for the sake of parsimony, we speciﬁed a model without this path: MLR χ2 (3, N = 126) = 5.14, P = .16, CFI =.97; RMSEA =.08; SRMR =.03. We conducted a nested model test according to L. K. Muthe´n and B. O.