Sample Size Planning for Power for Polynomial Change Models

Returns power given the sample size, or sample size given the desired power, for the group difference in a polynomial change coefficient (a flat-line intercept, a linear slope, a quadratic acceleration, or any higher-order trend) in a two-group longitudinal design, following Raudenbush and Liu (2001). The trend whose group difference is tested is selected with trend; trend = "linear" (the default) reproduces the straight-line case.

Usage

ss_power_pcm(
  beta,
  tau,
  level_1_variance,
  frequency,
  duration,
  desired_power = NULL,
  N = NULL,
  alpha_level = 0.05,
  standardized = TRUE,
  directional = FALSE,
  trend = "linear"
)

Arguments

beta

The level two regression coefficient for the group by time interaction in the polynomial change coefficient selected by trend (linear by default), where the grouping variable is coded -.5 and .5 for the two groups. When standardized = TRUE (the default) this is the standardized change difference (see standardized).

tau

The true between-subject variance of the individuals' change coefficient for the trend selected by trend (the variance of the slopes for trend = "linear").

level_1_variance

Level one (within-subject) error variance

frequency

Number of measurements per unit of time, where the unit is the one in which duration is expressed. It need not be a whole number; for example, frequency = 0.5 means one measurement every two time units. Together with duration it fixes the number of equally spaced measurement occasions, \(M = f \times D + 1\), with \(f\) the frequency and \(D\) the duration.

duration

Length of the study in the chosen time unit (for example, years, grades, or hours). Measurements are taken at \(M = f \times D + 1\) equally spaced occasions spanning time 0 to duration.

desired_power

Desired power

N

Total sample size (one-half in each of the two groups)

alpha_level

Type I error rate

standardized

The standardized change difference is the unstandardized change difference divided by the square root of tau, the between-subject variance of the change coefficient. TRUE (the default) treats beta as already standardized; FALSE treats it as the raw (unstandardized) change difference.

directional

Should a one (TRUE) or two (FALSE) tailed test be performed.

trend

The polynomial change coefficient whose group difference is tested, given either as a name ("intercept", "linear", "quadratic", "cubic", "quartic", ...) or as a non-negative integer order (0 = intercept / flat line, 1 = linear, 2 = quadratic, ...). Defaults to "linear". The design must supply at least \(p + 1\) measurement occasions to estimate a degree-\(p\) trend.

Value

A data.frame (class dmar_tbl) with one row per reported quantity in a term / value layout: the per-group size per group (necessary_n_per_group, or specified_n_per_group when N is supplied) and the total (total_N); the achieved power (actual_power); the measurement schedule (freq, duration, measurement_occasions); the polynomial order of the tested change coefficient (polynomial_order, 0 = intercept, 1 = linear, 2 = quadratic, ...); the unstandardized and standardized change difference (unstd_coefficient, std_coefficient); the level one error variance (l1_error_var); the true and error variance of the change coefficient (true_var_of_slopes, error_var_of_slopes, whose names retain "slopes" from the linear case); the change-coefficient reliability (reliability); and the noncentrality parameter of the t test (noncentral_t_parm).

Details

The two groups each contain \(N / 2\) subjects measured on \(M = f \times D + 1\) equally spaced occasions. Each subject's degree-\(p\) polynomial change coefficient is estimated within subject; the test compares the two group means of that coefficient. The change coefficient is taken in the derivative-scaled metric (\(p!\) times the leading coefficient of \(t^p\)), the metric in which the Raudenbush and Liu (2001) constants apply.

The within-subject sampling variance of the estimated coefficient is (Raudenbush & Liu, 2001, p. 392) $$V = \sigma^2_e\, f^{2p}\,\frac{(M - p - 1)!}{K_p\,(M + p)!}, \qquad \frac{1}{K_p} = \frac{(2p)!\,(2p+1)!}{(p!)^2},$$ so that \(1/K_p = 1, 12, 720, 100800, \ldots\) for \(p = 0, 1, 2, 3, \ldots\). This \(V\) equals \((p!)^2\) times the variance of the ordinary least squares estimate of the coefficient of \(t^p\), reduces to \(\sigma^2_e / M\) at \(p = 0\) and to \(12\,\sigma^2_e f^2 / [M(M^2 - 1)]\) at \(p = 1\). The slope reliability is \(\tau / (\tau + V)\) (their Equation 15), the variance of the between-group difference is \(4(\tau + V)/N\) (their Equation 10), and the t test has \(N - 2\) degrees of freedom and noncentrality \(\sqrt{N\,\beta^2\,[\tau/(\tau + V)]/4}\) (their Equations 12 and 14). The linear case reproduces the National Youth Survey benchmark in their Tables 1 and 2; the general-\(p\) formula has been checked against the exact \((X'X)^{-1}\) variance and against an end-to-end Monte Carlo power study for the quadratic trend.

References

Kelley, K., & Maxwell, S. E. (2008). Sample size planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuutari, L. Bickman, & J. Brannen (Eds.), The Sage handbook of social research methods (pp. 166–192). Sage.

Kelley, K., & Rausch, J. R. (2011). Sample size planning for longitudinal models: Accuracy in parameter estimation for polynomial change parameters. Psychological Methods, 16(4), 391–405. doi:10.1037/a0023352

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapters 11, 15.)

Maxwell, S. E., Kelley, K., & Rausch, J. R. (2008). Sample size planning for statistical power and accuracy in parameter estimation. Annual Review of Psychology, 59, 537–563. doi:10.1146/annurev.psych.59.103006.093735

Raudenbush, S. W., & Liu, X.-F. (2001). Effects of study duration, frequency of observation, and sample size on power in studies of group differences in polynomial change. Psychological Methods, 6(4), 387–401. doi:10.1037/1082-989X.6.4.387

See also

design_consequences for what a chosen design delivers: power, the Type S (sign) and Type M (exaggeration) errors of the significance filter, and the expected confidence interval width.

Other sample size for power: power_fisher_exact(), ss_aipe_mixed_effects(), ss_power_R2(), ss_power_R2_sensitivity(), ss_power_c(), ss_power_c_ancova(), ss_power_composite_ancova(), ss_power_composite_ancova_2group(), ss_power_composite_anova(), ss_power_composite_factorial_ancova(), ss_power_composite_factorial_ancova_het(), ss_power_composite_factorial_anova(), ss_power_composite_sem(), ss_power_contrast(), ss_power_equivalence_c(), ss_power_factorial_ancova(), ss_power_factorial_anova(), ss_power_indirect_effect(), ss_power_mixed_effects(), ss_power_one_way_anova(), ss_power_r(), ss_power_rc(), ss_power_reg_coef(), ss_power_reg_coef_sensitivity(), ss_power_rm_anova(), ss_power_sc(), ss_power_sem(), ss_power_smd(), ss_power_split_plot_anova()

Author

Ken Kelley kkelley@nd.edu

Examples

# The examples reproduce the National Youth Survey illustration of
# Raudenbush and Liu (2001, p. 393). One observation per year
# (frequency = 1) over a four-year study (duration = 4) gives
# M = frequency * duration + 1 = 5 equally spaced occasions. The
# standardized slope difference is -0.40, the slope variance is
# tau = 0.003, and the level-one error variance is 0.0262, so the slope
# reliability is 0.53.

# (1) Power at a given sample size. With N = 238 (119 per group) the
#     design has power 0.61 (Raudenbush and Liu, 2001, Table 1, D = 4,
#     f = 1).
ss_power_pcm(beta = -.4, tau = .003, level_1_variance = .0262,
             frequency = 1, duration = 4, N = 238)
#>  term                  value  
#>  specified_n_per_group 119    
#>  total_N               238    
#>  actual_power          0.612  
#>  freq                  1      
#>  duration              4      
#>  measurement_occasions 5      
#>  polynomial_order      1      
#>  unstd_coefficient     -0.0219
#>  std_coefficient       -0.4   
#>  l1_error_var          0.0262 
#>  true_var_of_slopes    0.003  
#>  error_var_of_slopes   0.00262
#>  reliability           0.534  
#>  noncentral_t_parm     2.25   

# (2) Sample size for a target power. Solving the same design for 0.80
#     power returns N = 370 (185 per group), between their Table 2 cells
#     N = 300 (power 0.71) and N = 400 (power 0.83).
ss_power_pcm(beta = -.4, tau = .003, level_1_variance = .0262,
             frequency = 1, duration = 4, desired_power = .80)
#>  term                  value  
#>  necessary_n_per_group 185    
#>  total_N               370    
#>  actual_power          0.801  
#>  freq                  1      
#>  duration              4      
#>  measurement_occasions 5      
#>  polynomial_order      1      
#>  unstd_coefficient     -0.0219
#>  std_coefficient       -0.4   
#>  l1_error_var          0.0262 
#>  true_var_of_slopes    0.003  
#>  error_var_of_slopes   0.00262
#>  reliability           0.534  
#>  noncentral_t_parm     2.81   

# (3) Unstandardized slope. The unstandardized slope difference is
#     beta * sqrt(tau) = -0.40 * sqrt(0.003) = -0.0219. Passing it with
#     standardized = FALSE reproduces the power of 0.61 from example (1).
ss_power_pcm(beta = -.0219, tau = .003, level_1_variance = .0262,
             frequency = 1, duration = 4, N = 238, standardized = FALSE)
#>  term                  value  
#>  specified_n_per_group 119    
#>  total_N               238    
#>  actual_power          0.612  
#>  freq                  1      
#>  duration              4      
#>  measurement_occasions 5      
#>  polynomial_order      1      
#>  unstd_coefficient     -0.0219
#>  std_coefficient       -0.4   
#>  l1_error_var          0.0262 
#>  true_var_of_slopes    0.003  
#>  error_var_of_slopes   0.00262
#>  reliability           0.534  
#>  noncentral_t_parm     2.25   

# (4) Longer study, same number of occasions. Doubling the duration to
#     D = 8 while keeping M = 5 (so frequency = 0.5, one observation every
#     two years) raises the slope reliability to 0.82 and power to about
#     0.80. Spreading the same five occasions over a longer span sharply
#     increases power (Raudenbush & Liu, 2001, p. 393).
ss_power_pcm(beta = -.4, tau = .003, level_1_variance = .0262,
             frequency = .5, duration = 8, N = 238)
#>  term                  value   
#>  specified_n_per_group 119     
#>  total_N               238     
#>  actual_power          0.795   
#>  freq                  0.5     
#>  duration              8       
#>  measurement_occasions 5       
#>  polynomial_order      1       
#>  unstd_coefficient     -0.0219 
#>  std_coefficient       -0.4    
#>  l1_error_var          0.0262  
#>  true_var_of_slopes    0.003   
#>  error_var_of_slopes   0.000655
#>  reliability           0.821   
#>  noncentral_t_parm     2.8     

# (5) More frequent sampling over a shorter span, same occasions. Halving
#     the duration to D = 2 while keeping M = 5 (so frequency = 2) drops
#     the slope reliability to 0.22 and power to about 0.31. Sampling more
#     often over a shorter study does little for power (Raudenbush & Liu,
#     2001, p. 393).
ss_power_pcm(beta = -.4, tau = .003, level_1_variance = .0262,
             frequency = 2, duration = 2, N = 238)
#>  term                  value  
#>  specified_n_per_group 119    
#>  total_N               238    
#>  actual_power          0.305  
#>  freq                  2      
#>  duration              2      
#>  measurement_occasions 5      
#>  polynomial_order      1      
#>  unstd_coefficient     -0.0219
#>  std_coefficient       -0.4   
#>  l1_error_var          0.0262 
#>  true_var_of_slopes    0.003  
#>  error_var_of_slopes   0.0105 
#>  reliability           0.223  
#>  noncentral_t_parm     1.46   

# (6) One-sided test. A directional test of the base design places the
#     whole Type I error rate in the predicted tail, raising power from
#     0.61 to about 0.73.
ss_power_pcm(beta = -.4, tau = .003, level_1_variance = .0262,
             frequency = 1, duration = 4, N = 238, directional = TRUE)
#>  term                  value  
#>  specified_n_per_group 119    
#>  total_N               238    
#>  actual_power          0.727  
#>  freq                  1      
#>  duration              4      
#>  measurement_occasions 5      
#>  polynomial_order      1      
#>  unstd_coefficient     -0.0219
#>  std_coefficient       -0.4   
#>  l1_error_var          0.0262 
#>  true_var_of_slopes    0.003  
#>  error_var_of_slopes   0.00262
#>  reliability           0.534  
#>  noncentral_t_parm     2.25   

# (7) A higher-order trend. The same machinery plans power for the group
#     difference in any polynomial change coefficient. Here the target is
#     the quadratic trend (curvature / acceleration): with eight occasions
#     (frequency = 1, duration = 7), a between-subject quadratic-coefficient
#     variance tau = 0.002, level-one error variance 0.05, and a
#     standardized quadratic difference of 0.45, the design is planned for
#     0.80 power. A quadratic trend needs at least three occasions; a cubic
#     at least four (trend = "cubic" or trend = 3), and so on.
ss_power_pcm(beta = 0.45, tau = 0.002, level_1_variance = 0.05,
             frequency = 1, duration = 7, desired_power = .80,
             trend = "quadratic")
#>  term                  value  
#>  necessary_n_per_group 125    
#>  total_N               250    
#>  actual_power          0.801  
#>  freq                  1      
#>  duration              7      
#>  measurement_occasions 8      
#>  polynomial_order      2      
#>  unstd_coefficient     0.0201 
#>  std_coefficient       0.45   
#>  l1_error_var          0.05   
#>  true_var_of_slopes    0.002  
#>  error_var_of_slopes   0.00119
#>  reliability           0.627  
#>  noncentral_t_parm     2.82