Multiple Linear Regression vs. Mixed-effects linear model

[Harry Stern]

Hi,

I would love some insight on running linear regressions.

I am working with some pre-post intervention data and I would like to run a regression using change in physical function (SPPB; baseline-12 months) as my predictor variable and change in physical health-related quality of life (PCS; baseline-12 months) as my outcome variable while adjusting for age, sex, race, weight change (baseline-12 months), and treatment group (phone or newsletter). Initially, I just ran a multiple linear regression using "change" variables, i.e. change in physical function = 12 month SPPB score - baseline SPPB score and change in physical health-related quality of life = 12 month PCS score - baseline PCS score. However, I have recently learned that I may need to run mixed-effects linear models since I am working with pre-post intervention data, which multiple linear regression may not be suitable for.

My first question is: should I be running a mixed-effect linear model or is multiple linear regression fine for this analysis?

If I should be running a mixed-effect linear model, how do I include independent variables, follow-up contact condition (phone, newsletter), visit (baseline, 12 months), and contact condition by visit interaction as fixed effects and subject as a random effect, and adjust for age, sex, and race, in Stata? Is this an appropriate set-up of the mixed-effect linear model for evaluating the relationship between change in physical function (predictor) and change in physical health-related quality of life (outcome variable) from baseline to 12 months?

Here is the data I am working with:

* Example generated by -dataex-. To install: ssc install dataex
clear
input long pid byte(sppbscore0 sppbscore12) float(sppb_ch_12mo PCS_score1 PCS_score3 PCS_ch_12mo wt1_mean wt12mo_mean wt_ch_12mo sex_r age_screen race_wnw) long assignment_r
10002  5  6  1  50.7 57.58   6.880001     93.45        79 -14.449997 0 65.41 0 2
10005  4  7  3 39.31 47.52   8.209999      67.6     63.45 -4.1499977 0 72.72 0 1
10008 11 10 -1  57.4  57.4          0     68.05 66.350006  -1.699997 0 74.42 0 2
10011  9  7 -2 50.87 45.19      -5.68      86.5      82.9 -3.5999985 1 78.44 0 2
10012 11 11  0 47.36 41.03  -6.330002        92      79.6 -12.400002 0 67.15 0 2
10013  7  5 -2 41.81  49.8   7.989998       111      89.3 -21.699997 0 71.13 0 1
10015 10  8 -2 55.69 52.68  -3.009998     100.5      94.3  -6.199997 0 65.43 0 2
10017  4  6  2 42.87 53.94      11.07     95.05      91.5  -3.550003 1 80.66 0 2
10024 10  9 -1 50.92 59.59   8.670002     106.1    107.25  1.1500015 0 65.35 1 2
10025  7  8  1  47.1 51.93   4.830002    109.05      96.3     -12.75 1 73.17 0 1
10026 10 11  1 46.81 61.08      14.27      84.7      76.6  -8.099998 0 69.09 0 2
10031  9  7 -2 38.36 39.03   .6699982      94.6      87.4  -7.199997 0 70.95 0 1
10033  9  9  0  57.4 53.92  -3.480003  93.64999     90.75  -2.899994 1 70.81 0 2
10044  6  6  0 53.86 52.38 -1.4799995      82.6      72.3 -10.299995 1 78.11 0 2
10047 11  8 -3  59.6 58.38 -1.2199974     78.75      72.9  -5.849998 0 66.85 1 2
10048  8  8  0 54.64 54.64          0      84.3      84.1 -.20000458 0 70.77 1 1
10054  9  9  0 57.12 41.75     -15.37      72.8      67.1  -5.700005 0 65.81 0 2
10055 11 10 -1 58.38 47.94 -10.440002    111.85     108.3  -3.550003 1 65.28 0 1
10058  8  6 -2 43.45 46.68  3.2299995      93.8      89.8         -4 0  69.4 1 2
10060 10  9 -1 51.28    51 -.27999878        70     64.95  -5.050003 0 66.33 0 2
10063 10  6 -4 49.81 21.66     -28.15    102.15      98.8  -3.349991 1 68.62 0 1
10066 11 11  0  53.5  53.5          0      89.8      90.1   .2999954 0 69.22 0 1
10068 12 11 -1 53.53  53.7    .170002     101.2      94.5  -6.699997 1 66.28 0 1
10069 10  9 -1 43.74 56.13      12.39      71.9      66.6  -5.300003 0 71.99 0 1
10071 10 11  1 42.75 53.33  10.580002       131     106.8 -24.199997 1 66.11 0 2
10077  7  6 -1 48.52  52.3   3.779999      94.7      86.2       -8.5 0 68.39 0 2
10079 10 10  0 52.51 58.89   6.380001 73.649994      66.1  -7.549995 0 72.74 0 2
10081  7  8  1 46.95 49.04  2.0900002      90.7      82.8  -7.899994 0 71.04 1 1
10085 10  8 -2 54.91 58.62   3.709999      85.2  79.85001  -5.349991 0 68.05 1 1
10087  6  7  1  26.1 22.69      -3.41     89.95      75.8 -14.149994 0 73.76 0 1
end
label values sex_r Sex
label def Sex 0 "Female", modify
label def Sex 1 "Male", modify
label values race_wnw race
label def race 0 "White", modify
label def race 1 "Non-white", modify
label values assignment_r assignment_r
label def assignment_r 1 "Newsletter", modify
label def assignment_r 2 "Phone", modify

My original multiple linear regression code was this:

regress PCS_ch_12mo sppb_ch_12mo age_screen sex_r race_wnw assignment_r wt_ch_12mo

.

Another question, if the multiple linear regression is fine, should I include baseline PCS score as a covariate?

Thank you for any help.

[Clyde Schechter]

Change scores have the advantage of being easy to explain to non-technical audiences, who (think) they understand them. But their use is perilous unless a number of conditions are satisfied. Frank Harrell has a nice explanation of this at https://www.fharrell.com/post/errmed/#change. (In fact, read the whole piece when you get a chance--it's teaches a number of important lessons about many important errors that are very common in medical research. It's well written and not that long.) Anyway, I would say that your PCS scores are not really suitable for change-score analysis because looking at histograms, they seem to have a ceiling effect.

If your actual research goal is to assess the relationship between change in sppbscore and change in pcs_score, then this is very unfortunate, because it means your data are not suitable for purpose. But if your real goal here is to estimate a treatment effect (of assignment_r) and you simply want to adjust for the effect of sppbscore on pcs_score, then a mixed model will do the trick.

[Harry Stern]

Change scores have the advantage of being easy to explain to non-technical audiences, who (think) they understand them. But their use is perilous unless a number of conditions are satisfied. Frank Harrell has a nice explanation of this at https://www.fharrell.com/post/errmed/#change. (In fact, read the whole piece when you get a chance--it's teaches a number of important lessons about many important errors that are very common in medical research. It's well written and not that long.) Anyway, I would say that your PCS scores are not really suitable for change-score analysis because looking at histograms, they seem to have a ceiling effect.

If your actual research goal is to assess the relationship between change in sppbscore and change in pcs_score, then this is very unfortunate, because it means your data are not suitable for purpose. But if your real goal here is to estimate a treatment effect (of assignment_r) and you simply want to adjust for the effect of sppbscore on pcs_score, then a mixed model will do the trick.

I see. The range of PCS_score is 0-100 and the range for my sample is about 20-60 for PCS_score1 (pre-intervention) and PCS_score3 (post-intervention) with a mean of ~50 for both scores, so is there a ceiling effect?

My research question is to see how change in sppbscore predicts change in PCS_score from pre-post intervention (i.e. baseline-12 months).

I am including more data points below - maybe you will not see a ceiling effect when more subjects are included?

* Example generated by -dataex-. To install: ssc install dataex
clear
input long pid byte(sppbscore0 sppbscore12) float(sppb_ch_12mo PCS_score1 PCS_score3 PCS_ch_12mo wt1_mean wt12mo_mean wt_ch_12mo sex_r age_screen race_wnw) long assignment_r
10002  5  6  1  50.7 57.58   6.880001     93.45        79 -14.449997 0 65.41 0 2
10005  4  7  3 39.31 47.52   8.209999      67.6     63.45 -4.1499977 0 72.72 0 1
10008 11 10 -1  57.4  57.4          0     68.05 66.350006  -1.699997 0 74.42 0 2
10011  9  7 -2 50.87 45.19      -5.68      86.5      82.9 -3.5999985 1 78.44 0 2
10012 11 11  0 47.36 41.03  -6.330002        92      79.6 -12.400002 0 67.15 0 2
10013  7  5 -2 41.81  49.8   7.989998       111      89.3 -21.699997 0 71.13 0 1
10015 10  8 -2 55.69 52.68  -3.009998     100.5      94.3  -6.199997 0 65.43 0 2
10017  4  6  2 42.87 53.94      11.07     95.05      91.5  -3.550003 1 80.66 0 2
10024 10  9 -1 50.92 59.59   8.670002     106.1    107.25  1.1500015 0 65.35 1 2
10025  7  8  1  47.1 51.93   4.830002    109.05      96.3     -12.75 1 73.17 0 1
10026 10 11  1 46.81 61.08      14.27      84.7      76.6  -8.099998 0 69.09 0 2
10031  9  7 -2 38.36 39.03   .6699982      94.6      87.4  -7.199997 0 70.95 0 1
10033  9  9  0  57.4 53.92  -3.480003  93.64999     90.75  -2.899994 1 70.81 0 2
10044  6  6  0 53.86 52.38 -1.4799995      82.6      72.3 -10.299995 1 78.11 0 2
10047 11  8 -3  59.6 58.38 -1.2199974     78.75      72.9  -5.849998 0 66.85 1 2
10048  8  8  0 54.64 54.64          0      84.3      84.1 -.20000458 0 70.77 1 1
10054  9  9  0 57.12 41.75     -15.37      72.8      67.1  -5.700005 0 65.81 0 2
10055 11 10 -1 58.38 47.94 -10.440002    111.85     108.3  -3.550003 1 65.28 0 1
10058  8  6 -2 43.45 46.68  3.2299995      93.8      89.8         -4 0  69.4 1 2
10060 10  9 -1 51.28    51 -.27999878        70     64.95  -5.050003 0 66.33 0 2
10063 10  6 -4 49.81 21.66     -28.15    102.15      98.8  -3.349991 1 68.62 0 1
10066 11 11  0  53.5  53.5          0      89.8      90.1   .2999954 0 69.22 0 1
10068 12 11 -1 53.53  53.7    .170002     101.2      94.5  -6.699997 1 66.28 0 1
10069 10  9 -1 43.74 56.13      12.39      71.9      66.6  -5.300003 0 71.99 0 1
10071 10 11  1 42.75 53.33  10.580002       131     106.8 -24.199997 1 66.11 0 2
10077  7  6 -1 48.52  52.3   3.779999      94.7      86.2       -8.5 0 68.39 0 2
10079 10 10  0 52.51 58.89   6.380001 73.649994      66.1  -7.549995 0 72.74 0 2
10081  7  8  1 46.95 49.04  2.0900002      90.7      82.8  -7.899994 0 71.04 1 1
10085 10  8 -2 54.91 58.62   3.709999      85.2  79.85001  -5.349991 0 68.05 1 1
10087  6  7  1  26.1 22.69      -3.41     89.95      75.8 -14.149994 0 73.76 0 1
10089 11 11  0 53.58 53.64  .05999756     119.7     122.4  2.7000046 1 68.62 0 1
10091  8  9  1    32    41          9      69.4      66.5 -2.9000015 0 72.87 0 2
10092  4  3 -1 52.51 55.27   2.760002      91.9      89.4       -2.5 0 73.45 0 1
10095  7  6 -1 53.81 52.76  -1.050003      88.3      81.9  -6.400002 0  65.1 0 1
10098  9 10  1 42.68 52.77      10.09  89.64999      88.9  -.7499924 0 68.94 0 2
10099  7 11  4 52.29 54.91   2.619999     84.55      77.9  -6.650002 0 67.33 0 1
10100 11 10 -1 49.15 48.42  -.7300034      70.3      72.2   1.899994 0 67.05 0 1
10101 12 12  0 42.84    51       8.16     89.55      82.6  -6.950005 0 64.94 0 2
10105  7  8  1  53.5 53.78  .27999878      78.4     71.45  -6.950005 0 76.11 0 2
10106 11 10 -1 58.32 54.07      -4.25      86.4        86  -.4000015 0 65.46 0 1
10108  7  8  1 58.38 54.75  -3.630001      95.6      88.3  -7.299995 1 79.98 0 1
10113  6  7  1 47.87 50.03  2.1599998    146.05     142.6  -3.449997 1 70.15 0 2
10116  7  7  0 47.22 42.13      -5.09  99.85001      99.5  -.3500061 0 69.17 0 2
10117  7  6 -1 52.51 33.14     -19.37     93.45      95.1  1.6500015 0 71.58 0 1
10119  8  7 -1 47.84 51.41       3.57      76.7      72.4 -4.2999954 0 67.84 0 1
10121 10 11  1 51.35 56.36   5.010002      73.5      68.7  -4.800003 0 67.14 0 1
10122 11 12  1  48.7 59.87  11.169998      85.3      75.7  -9.600006 1 69.14 0 2
10124  8  7 -1 56.61 57.12   .5099983     73.95     66.45       -7.5 0 72.97 0 2
10125  6  7  1  46.1 40.64  -5.459999    102.95     102.3  -.6499939 0 70.77 0 1
20002  8  8  0 51.49 45.17  -6.320004  89.64999      78.7 -10.949997 1 73.99 0 1
20004 12 11 -1 38.55 42.56   4.010002     105.6      99.7  -5.900002 1 67.95 0 1
20005  6  7  1 46.08     .          .      94.5     99.95   5.449997 0 67.73 0 2
20006 10 10  0  42.9 54.54      11.64      67.1      61.6       -5.5 0 72.38 0 1
20009  9 10  1 50.79 51.11   .3199997     85.45      81.6 -3.8499985 1 73.25 0 1
20010  9 11  2 52.99 54.76  1.7699966      85.7        75 -10.699997 1 77.24 0 2
20011 10  9 -1 53.09 56.63   3.540001      90.4     80.95  -9.450005 0 66.37 0 2
20014  8 10  2 47.22  37.3  -9.920002      87.3      90.6  3.2999954 0 66.89 0 1
20017  7  5 -2  27.1 24.24 -2.8600006        92      88.9 -3.0999985 1 73.93 0 1
20018  6  7  1 47.94 53.78       5.84      80.2        75  -5.199997 0 78.95 0 2
20019  8  7 -1 38.79 49.66      10.87      77.2      75.1 -2.0999985 0 70.58 0 1
20021 12 12  0 59.16 61.42  2.2599983      93.9        79 -14.900002 1 65.45 1 2
20022  7  9  2  43.9 49.23   5.329998      89.7      81.8  -7.899994 0 65.16 0 2
20023  8  9  1 45.46 55.63  10.170002      80.9      67.1 -13.800003 0 65.17 0 2
20025  9  9  0 52.29     .          .     109.2     109.8   .6000061 0 67.28 0 2
20026 12 12  0 57.12 57.12          0      80.2      76.3  -3.899994 1 65.43 0 1
20028 11 10 -1 53.37    59   5.630001      76.6     74.75 -1.8499985 0 73.25 1 1
20030 12 11 -1 59.38 58.11 -1.2700005      67.7      71.2        3.5 0 67.64 1 1
20034  5  8  3 54.64 53.04 -1.5999985      66.9 64.850006 -2.0499954 0 69.82 0 1
20035 11 11  0 56.03 53.23  -2.799999      82.4     77.55  -4.849998 0  69.2 0 2
20037 10  9 -1 54.25 57.12   2.869999     105.1      95.2  -9.900002 1  70.4 0 2
20039  8  5 -3 51.17     .          .      79.5      76.9 -2.5999985 0    73 0 2
20042  6  .  .  37.8     .          .    113.95         .          . 1 75.89 0 2
20043 10 10  0 57.12 61.64  4.5200005      84.4      80.1  -4.300003 0 68.19 0 1
20046  9 10  1  55.9 56.83   .9300003     142.5     132.5        -10 0 67.29 0 1
20051  8  9  1 58.34 52.99  -5.349998     95.05      94.1  -.9500046 0  74.9 0 2
20054  9 11  2 55.63 56.45   .8199997      74.1     72.75 -1.3499985 0 69.19 0 1
20058  3  4  1 47.53 43.01 -4.5200005    121.35     118.5  -2.850006 1 77.73 0 2
20060  8  .  . 53.78     .          .      83.6         .          . 0 70.93 0 2
20063  5  7  2 48.52    50  1.4799995      97.3        92  -5.300003 0 76.52 0 1
20070 10 11  1 57.62 57.62          0      80.6      78.6         -2 0 66.05 0 2
20073 10  7 -3 55.24 58.38  3.1399994      96.2      95.5  -.6999969 0 66.35 0 1
20074  8 10  2 47.25     .          .      82.5      74.4  -8.099998 0  66.8 0 1
20076 11  8 -3 52.18 35.95     -16.23      78.2 75.850006 -2.3499908 0 73.07 0 1
20080  7  6 -1 41.35 36.48  -4.869999      83.3      74.4  -8.900002 0 73.36 0 2
20081 11 10 -1 48.31 52.84   4.529999      84.5  80.35001  -4.149994 1 73.82 0 1
20082  6  6  0 50.78 62.36  11.580002      85.5  82.64999  -2.850006 0 75.59 0 2
20085  5  .  . 51.03     .          .     105.9         .          . 0 75.43 0 1
20087  8  9  1 44.61 47.68       3.07 69.649994      62.2  -7.449993 0 67.97 0 2
20092  7  8  1 39.62 34.55      -5.07  77.35001      70.1  -7.250008 0  73.9 0 1
20093  8  9  1 48.04 56.48   8.439999      85.9     85.45  -.4500046 1 65.39 0 1
20094  7  7  0 43.11 46.04       2.93      89.3      78.7 -10.600006 0 79.01 1 1
20095 12 12  0 57.96 61.93   3.970001     104.7      94.8  -9.899994 1 66.05 0 2
20096 11  .  .  57.4     .          .      89.8         .          . 0 66.99 0 2
20097  8  9  1 58.38     .          .     88.25      94.2   5.949997 0 67.02 0 1
20098  8  8  0 56.68 61.79   5.110001     104.8     103.7 -1.1000061 0 68.82 0 2
20099  8 11  3 47.62 47.19  -.4300003      88.9      78.8 -10.099998 1  71.4 0 1
20100  9  8 -1 54.36 44.21 -10.150002     103.2     103.2          0 1 67.15 0 2
20103  5  7  2 52.45 52.96   .5099983    123.25     121.8  -1.449997 0 76.98 0 2
20107  7  6 -1 52.87 57.12       4.25     100.6     96.25 -4.3499985 1 76.97 0 1
20108  9  7 -2  53.5  59.1   5.599998      77.4        84   6.599998 0  66.8 0 1
20110  7  7  0 49.43  56.4   6.970001  87.35001     79.55  -7.800003 0 75.88 0 2
20113 12 11 -1 55.52 57.12  1.5999985      72.4      64.6  -7.800003 0 71.59 0 1
20114  8  9  1 29.59 51.43      21.84      79.4      64.4        -15 0 79.34 0 1
20115 10 11  1 55.52 46.02       -9.5      70.8      65.7  -5.100006 0  65.4 0 1
20116  8  6 -2  36.2 39.26  3.0599976      71.3 66.350006  -4.949997 0 74.81 0 2
20117  6  6  0 45.65 54.35   8.699997     107.8      99.8         -8 0 76.49 0 2
20118  6  5 -1 45.46  42.6 -2.8600006      88.7      86.2       -2.5 0 76.11 0 2
20123  5  7  2 60.91 58.38  -2.529999      81.1        75  -6.099998 0 79.54 0 1
20124  8  9  1 52.48 55.82       3.34      84.2      80.1 -4.0999985 0 74.98 0 1
20125  5  7  2 55.62 57.18  1.5600014      66.8      56.6 -10.200005 0 76.85 0 2
20128  7 11  4 57.12 54.25  -2.869999      78.3      68.8       -9.5 0 69.08 0 2
20129  5  6  1 36.52 47.35  10.829998    100.65      94.2  -6.449997 0 79.66 0 1
20130  5  6  1 46.71 50.93   4.220001    107.35     101.5  -5.850006 0 79.11 0 2
20131 11  9 -2 45.88 48.62   2.739998      63.2      66.7   3.499996 0  74.3 0 1
20136  7  8  1 55.04 57.19  2.1499977        77      71.5       -5.5 0 74.75 0 2
20137  7  4 -3 42.18 28.42     -13.76      93.1     87.05  -6.049995 0 70.15 0 2
20139  5  9  4 32.89 53.37      20.48     105.6     103.4  -2.199997 1 68.28 0 2
20142  8  9  1 50.81 55.47       4.66     86.25      85.9  -.3499985 1 66.97 0 1
20154 10  9 -1 51.28 58.89   7.610001      74.8      62.8 -12.000004 0 67.51 0 2
20156  6  7  1  49.5 56.52       7.02      81.6        76  -5.599998 0 71.33 0 2
20161  4  5  1 44.97 46.09  1.1199989      94.7      91.4 -3.2999954 0  78.1 0 1
20164  9  7 -2 57.34  58.6  1.2599983      91.1      76.9 -14.199997 1 73.59 0 1
20166 10  9 -1 58.89 59.68   .7900009      83.7      71.4 -12.299995 1 71.76 0 1
20167  9  8 -1 43.25    44        .75     73.55      68.5  -5.050003 0 72.15 0 1
20169  8  9  1 54.16    51      -3.16     110.1     101.4  -8.699997 1 66.44 0 2
end
label values sex_r Sex
label def Sex 0 "Female", modify
label def Sex 1 "Male", modify
label values race_wnw race
label def race 0 "White", modify
label def race 1 "Non-white", modify
label values assignment_r assignment_r
label def assignment_r 1 "Newsletter", modify
label def assignment_r 2 "Phone", modify

[Clyde Schechter]

With the additional data, it doesn't look as bad as before. The histograms for the PCS scores are peaking close to the right end of the graph, but there is a bit of fall-off before the very right end is reached. It's certainly not ideal, and I would be very cautious in interpreting any analysis based on the PCS score change as a result. The bias that you have to deal with in using a change score with this kind of data is that the PCS measure apparently does not register degrees of health-related quality of life that are higher than 60, but the distributions suggest that there may be an appreciable number of people who would have such scores were it possible to measure them. If their quality of life changes within that higher than 60 range, the PCS measure will register no change at all. For people who start out just above 60 and it goes below, the change will be underestimated. (Think of a scale that tops out at 300 lbs--they are common in the real world. If a person loses weight from 350 lbs to 330, that scale shows zero change. A person starting at 320 pounds who gets down to 290 will be said to have lost only 10 pounds instead of the actual 30.)

But it's better than the first look, where my sense was that it is hopeless.

[Harry Stern]

With the additional data, it doesn't look as bad as before. The histograms for the PCS scores are peaking close to the right end of the graph, but there is a bit of fall-off before the very right end is reached. It's certainly not ideal, and I would be very cautious in interpreting any analysis based on the PCS score change as a result. The bias that you have to deal with in using a change score with this kind of data is that the PCS measure apparently does not register degrees of health-related quality of life that are higher than 60, but the distributions suggest that there may be an appreciable number of people who would have such scores were it possible to measure them. If their quality of life changes within that higher than 60 range, the PCS measure will register no change at all. For people who start out just above 60 and it goes below, the change will be underestimated. (Think of a scale that tops out at 300 lbs--they are common in the real world. If a person loses weight from 350 lbs to 330, that scale shows zero change. A person starting at 320 pounds who gets down to 290 will be said to have lost only 10 pounds instead of the actual 30.)

But it's better than the first look, where my sense was that it is hopeless.

Ah, I see. That is a very helpful explanation.

One more thought, The PCS scale was designed so that in normative data the mean score is 50 with a score of >50 indicating better physical health-related quality of life (HRQoL) than the average and worse physical HRQoL for a score <50. Does that change your concern that the PCS measure has a ceiling effect? Also, in this particular sample, I would not expect to see many high PCS scores, i.e. >60, so that may be contributing to the distribution of the data?

[Clyde Schechter]

The PCS scale was designed so that in normative data the mean score is 50 with a score of >50 indicating better physical health-related quality of life (HRQoL) than the average and worse physical HRQoL for a score <50. Does that change your concern that the PCS measure has a ceiling effect? Also, in this particular sample, I would not expect to see many high PCS scores, i.e. >60, so that may be contributing to the distribution of the data?

Perhaps. If the top score in your data being 60 represents your choice of a study population in which scores above that level just don't happen much or at all, then it's OK to use this variable for a change score. I was operating on the assumption that the maximum score of 60 was a limitation of the PCS scale itself and that the bunching up of scores near the top end meant that the PCS measurement was incapable of distinguishing degrees of health-related quality of life greater than that.

But if you have not done so yet, do read that section on change scores in the Harrell reference I linked to in #2. There are other potential problems with change scores: review your situation to see if any of those apply to you, and if so, whether you can find a way to fix it. In particular, if the absence of scores above 60 is because the inclusion/exclusion criteria for your study required a score of 60 or less, or imposed some other restriction that more or less implies that the PCS score will be 60 or less, then you will have a regression to the mean problem, unless the PCS_score1 in your data is a re-assessment, not the one that was used to determine eligibility for the study.

[Harry Stern]

Thank you for your help, Clyde. I did look at the Harrell reference and I will think it about more in the context of the data I am using.