Results

We first present results from the sip size and difficulty measures separately. Next, we report the relationship between the two measures.

Sip size

Figure 1. Sip size (ml) plotted by sitting angle. Color shows the number of pills; subplots show the two utensils. Points are jittered slightly in the horizontal direction to avoid overplotting. Lines show best linear fit with 95\% confidence intervals shaded.

Figure 1. Sip size (ml) plotted by sitting angle. Color shows the number of pills; subplots show the two utensils. Points are jittered slightly in the horizontal direction to avoid overplotting. Lines show best linear fit with 95% confidence intervals shaded.

Figure 1 shows sip size, plotted by angle, number of pills, and utensil. Descriptively, sip size was overall larger for cups rather than straws, larger for greater numbers of pills, and larger when patients were more upright (greater angle).

d$angle_centered <- as.numeric(d$angle) - 65
d$pills_centered <- as.numeric(d$pills) - 1
sip_mod <- lmer(sip ~ angle_centered * pills_centered * utensil + (1 | ID), 
     data = d)

sip_apa <- papaja::apa_print(sip_mod)
knitr::kable(summary(sip_mod)[[10]], digits = 3, caption = "All coefficients for mixed effects model table.")
All coefficients for mixed effects model table.
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 25.275 1.047 37.179 24.133 0.000
angle_centered 0.092 0.026 418.000 3.495 0.001
pills_centered 6.667 0.800 418.000 8.335 0.000
utensilStraw -3.499 0.924 418.000 -3.788 0.000
angle_centered:pills_centered -0.054 0.032 418.000 -1.661 0.097
angle_centered:utensilStraw 0.027 0.037 418.000 0.723 0.470
pills_centered:utensilStraw -2.750 1.131 418.000 -2.431 0.015
angle_centered:pills_centered:utensilStraw -0.049 0.046 418.000 -1.078 0.282

To quantify these trends, we fit a linear mixed effects model. We predicted sip size (in ml) as a function of bed angle (centered), number of pills (centered), and utensil (straw or cup), as well as all two- and three-way interactions of these variables. We included a random intercept for each participant. The model was fit with the lme4 package in R (Bates D, Mächler M, Bolker B, Walker S (2015). “Fitting Linear Mixed-Effects Models Using lme4.” Journal of Statistical Software, 67(1), 1–48. doi: 10.18637/jss.v067.i01.) and computed p-values with the lmerTest package.

Confirming our visual impression, higher angles led to significantly larger sip sizes (\(\hat{\beta} = 0.09\), 95% CI \([0.04, 0.14]\), \(t(418) = 3.50\), \(p = .001\)). Larger numbers of pills similarly led to larger sip sizes (\(\hat{\beta} = 6.67\), 95% CI \([5.10, 8.23]\), \(t(418) = 8.33\), \(p < .001\)). The use of a straw led to significantly smaller sip sizes (\(\hat{\beta} = -3.50\), 95% CI \([-5.31, -1.69]\), \(t(418) = -3.79\), \(p < .001\)). There was a trend towards an interaction of angle and number of pills (\(\hat{\beta} = -0.05\), 95% CI \([-0.12, 0.01]\), \(t(418) = -1.66\), \(p = .097\)), and a significant interaction of number of pills and the use of a straw (\(\hat{\beta} = -2.75\), 95% CI \([-4.97, -0.53]\), \(t(418) = -2.43\), \(p = .015\)).

Are there outliers?

Subject C1-011 is very close to being an outlier - not 3SDs above the mean, but definitely > 2 SDs above the mean in average sip size. Let’s just rerun everything without them.

Figure 1. Sip size (ml) plotted by sitting angle. Color shows the number of pills; subplots show the two utensils. Points are jittered slightly in the horizontal direction to avoid overplotting. Lines show best linear fit with 95\% confidence intervals shaded.

Figure 1. Sip size (ml) plotted by sitting angle. Color shows the number of pills; subplots show the two utensils. Points are jittered slightly in the horizontal direction to avoid overplotting. Lines show best linear fit with 95% confidence intervals shaded.

Figure 1 shows sip size, plotted by angle, number of pills, and utensil. Descriptively, sip size was overall larger for cups rather than straws, larger for greater numbers of pills, and larger when patients were more upright (greater angle).

sip_mod <- lmer(sip ~ angle_centered * pills_centered * utensil + (1 | ID), 
                data = filter(d, ID != "C1-011"))

sip_apa <- papaja::apa_print(sip_mod)
knitr::kable(summary(sip_mod)[[10]], digits = 3, caption = "All coefficients for mixed effects model table.")
All coefficients for mixed effects model table.
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 24.543 0.902 40.992 27.200 0.000
angle_centered 0.071 0.026 401.000 2.789 0.006
pills_centered 6.189 0.780 401.000 7.937 0.000
utensilStraw -3.263 0.900 401.000 -3.624 0.000
angle_centered:pills_centered -0.066 0.031 401.000 -2.091 0.037
angle_centered:utensilStraw 0.039 0.036 401.000 1.082 0.280
pills_centered:utensilStraw -2.508 1.103 401.000 -2.275 0.023
angle_centered:pills_centered:utensilStraw -0.056 0.044 401.000 -1.260 0.208

To quantify these trends, we fit a linear mixed effects model. We predicted sip size (in ml) as a function of bed angle (centered), number of pills (centered), and utensil (straw or cup), as well as all two- and three-way interactions of these variables. We included a random intercept for each participant. The model was fit with the lme4 package in R (Bates D, Mächler M, Bolker B, Walker S (2015). “Fitting Linear Mixed-Effects Models Using lme4.” Journal of Statistical Software, 67(1), 1–48. doi: 10.18637/jss.v067.i01.) and computed p-values with the lmerTest package.

Confirming our visual impression, higher angles led to significantly larger sip sizes (\(\hat{\beta} = 0.07\), 95% CI \([0.02, 0.12]\), \(t(401.00) = 2.79\), \(p = .006\)). Larger numbers of pills similarly led to larger sip sizes (\(\hat{\beta} = 6.19\), 95% CI \([4.66, 7.72]\), \(t(401.00) = 7.94\), \(p < .001\)). The use of a straw led to significantly smaller sip sizes (\(\hat{\beta} = -3.26\), 95% CI \([-5.03, -1.50]\), \(t(401.00) = -3.62\), \(p < .001\)). There was a trend towards an interaction of angle and number of pills (\(\hat{\beta} = -0.07\), 95% CI \([-0.13, 0.00]\), \(t(401.00) = -2.09\), \(p = .037\)), and a significant interaction of number of pills and the use of a straw (\(\hat{\beta} = -2.51\), 95% CI \([-4.67, -0.35]\), \(t(401.00) = -2.27\), \(p = .023\)).

Rating data

Figure 2. Rating data plotted by sitting angle. Plotting conventions are as in Figure 1.

Figure 2. Rating data plotted by sitting angle. Plotting conventions are as in Figure 1. 

likert_mod <- lmer(rating ~ angle_centered * pills_centered * utensil + (1 | ID), 
     data = d)

rating_apa <- papaja::apa_print(likert_mod)

Figure 2 shows the rating of difficulty across conditions. Swallowing more pills was rated as more difficult, and difficulty decreased with angle. Overall, swallowing the pills appeared to be more difficult in the straw condition.

To quantify these trends we fit an identical model to the above, except this time predicting likert ratings rather than sip sizes. Higher angles led to significantly lower difficulty ratings (\(\hat{\beta} = -0.02\), 95% CI \([-0.03, -0.02]\), \(t(418.00) = -10.34\), \(p < .001\)). Larger numbers of pills led to significantly higher difficulty ratings (\(\hat{\beta} = 0.72\), 95% CI \([0.59, 0.85]\), \(t(418.00) = 10.84\), \(p < .001\)). The use of a straw led to significantly lower difficulties overall (\(\hat{\beta} = -0.04\), 95% CI \([-0.19, 0.11]\), \(t(418.00) = -0.56\), \(p = .574\)), but there was a significant positive interaction between number of pills and the use of a straw (\(\hat{\beta} = 0.23\), 95% CI \([0.04, 0.41]\), \(t(418.00) = 2.42\), \(p = .016\)), suggesting that swallowing larger numbers of pills was substantially harder with a straw (across angles).

knitr::kable(summary(likert_mod)[[10]], digits = 3, caption = "All coefficients for mixed effects model table.")
All coefficients for mixed effects model table.
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 1.223 0.070 48.502 17.371 0.000
angle_centered -0.023 0.002 418.000 -10.339 0.000
pills_centered 0.717 0.066 418.000 10.835 0.000
utensilStraw -0.043 0.076 418.000 -0.563 0.574
angle_centered:pills_centered 0.003 0.003 418.000 1.211 0.227
angle_centered:utensilStraw -0.001 0.003 418.000 -0.298 0.766
pills_centered:utensilStraw 0.226 0.094 418.000 2.416 0.016
angle_centered:pills_centered:utensilStraw -0.006 0.004 418.000 -1.625 0.105

Correlational analysis