Descriptive Analysis
“A picture is worth a thousand words.” In our case, we are interested in whether the changes of blood pressure from the baseline to the dialysis follow-ups are different in cold and warm groups. To simplify our question, we only focus on the changes from the pre-dialysis to the first and second hours during the dialysis sessions. Also, by assuming all missing values are missing completely random, we can ignore these missing values (native method).
temp Mean (Systolic) Mean (Diastolic) Median (Systolic) Median (Diastolic) sd (Systolic) sd (Diastolic)
hot 15.24 7.31 15.00 6.00 18.58 13.00
cold 13.72 4.94 14.00 5.00 17.06 10.69
The above table summarizes the changes of blood pressure (diastolic/systolic) from the pre-dialysis to the first hour during the dialysis sessions in cold and hot (warm) groups. Overall, the median of the changes of blood pressure do not have huge differences in cold and hot (warm) groups. However, it seems that the mean and variation in cold group are smaller than those in hot (warm) group in diastolic. Therefore, two treatments (hot and cold) may not have a huge difference.
temp period Mean of Difference (Systolic) Median (Systolic) sd (Systolic)
hot 1.00 12.25 10.00 21.85
hot 2.00 16.05 16.00 17.55
cold 1.00 14.59 14.00 18.52
cold 2.00 13.36 14.00 16.42
temp period Mean of Difference (Diastolic) Median (Diastolic) sd (Diastolic)
hot 1.00 4.26 3.00 12.80
hot 2.00 8.13 7.00 12.94
cold 1.00 5.52 5.00 12.36
cold 2.00 4.69 5.00 9.92
Since we have a crossover study design, whether the participants were in warm or cold group in period 1 does matter in our analyses. The above two figures and tables summarize the changes of blood pressure (diastolic/systolic) from the pre-dialysis to the first hour during the dialysis sessions in cold and hot(warm) groups through separating participants by period. In terms of systolic, if participants were in warm group first, they would have slightly higher changes of blood pressure in cold group (period 2). However, patients would have higher changes of blood pressure in period 2 if they were in cold group in period 1. This is similar to diastolic. In this case, period may play a role in predicting the changes of blood pressure.
temp baseline (Systolic) baseline (Diastolic) Mean (Systolic 1) Mean (Diastolic 1) Mean (Systolic 2) Mean (Diastolic 2)
hot 145.44 81.42 130.58 74.20 129.69 74.31
cold 148.36 81.50 134.62 76.55 135.81 78.14
The above two figures and tables summarize the changes of blood pressure (diastolic/systolic) from the pre-dialysis to the first two hours during the dialysis sessions in cold and hot(warm) groups. It can be noticed that the blood pressure is decreasing from baseline to the first hour for all treatments in both systolic and diastolic. However, when we consider the second hour, the results have changed. The blood pressure is increasing in cold group and keeping the same in hot group (in both systolic and diastolic). Due to time limitation, we only do the confirmative analyses for the first hour. This is one limitation for this report, which can be further improved.
Preliminary Analysis
To help to confirm what we have seen in above section, we can analyze our data by following the Hill's and Armitage approach, which can be found here. In order to use this method, each participants can only have two paired measures (one for hot group and one for cold group). Thus, we took the average of the changes of blood pressure for each patients in every group. The four tables below are summarized statistics based on the Hill's and Armitage approach.
Group A (hot followed by cold) Diastolic
Period 1 Cold (Dias) Period 2 Hot (Dias) (1) - (2) ((1) + (2))/2 H - C
mean 3.96 7.79 -3.83 5.88 3.83
sd 6.83 6.83 5.82 6.18 5.82
Group B (cold followed by hot) Diastolic
Period 1 Hot (Dias) Period 2 Cold (Dias) (1) - (2) ((1) + (2))/2 H - C
mean 4.82 4.38 0.44 4.60 0.44
sd 7.60 4.11 8.21 4.53 8.21
Group A (hot followed by cold) Systolic
Period 1 Cold (Sys) Period 2 Hot (Sys) (1) - (2) ((1) + (2))/2 H - C
mean 9.76 15.82 -6.05 12.79 6.05
sd 13.22 9.99 10.99 10.35 10.99
Group B (cold followed by hot) Systolic
Period 1 Hot (Sys) Period 2 Cold (Sys) (1) - (2) ((1) + (2))/2 H - C
mean 11.31 12.99 -1.68 12.15 -1.68
sd 10.78 7.79 8.78 8.32 8.78
Based on the Hill's and Armitage's theory, we can investigate the possibility of a treatment effect, period effect, and treatment-period interaction through doing two sample t tests. The results of these t tests are summarized in the following table:
Systolic Diastolic
Treatment 0.15 0.06
Period 0.01 0.14
Treatment-Period Interaction 0.82 0.43
Above table shows that period is statistically significant in systolic, which is consistent with what we found in descriptive analysis. Thus, the corresponding p-value for treatment can be influenced by the significance of period. Therefore, in terms of treatment, the p-value in systolic may not accurate. Another interesting point is that the p-value of treatment in diastolic is just above 5% significant boundary. As I mentioned in early sections, we deleted several missing values by assuming they are missing at completely random. Therefore, the conclusion for treatment in diastolic can be easily altered if we do not have any missing values. In this situation, in order to draw more reliable conclusion, more analyses are required.
Analysis of Covariance Approach
Instead of using the Hill's and Armitage approach, we can further confirm our results by following an analysis of covariance approach (ANCOVA). You can find detailed information about ANCOVA here. Even though some people try to use change scores to fix the problem of chance imbalances at baseline between two treatment groups, analyzing change scores does not truly control for baseline imbalance problem. In particular, if participants have higher blood pressure at baseline, logically it is easier for them to decrease their blood pressure during dialysis. Therefore, ANCOVA is a better way to use. In order to use ANCOVA, we consider the baseline measurements, first hour measurements, period, temp, and the interaction of period and temp for both systolic and diastolic. We tried both linear models and mixed effect models with random effect on id. The results show that baseline, temp, and interaction are statistically significant at 5% level in both systolic and diastolic, which are totally different from what we have found in preliminary analyses. In general, compared with other methods, such as analysis of change scores, ANCOVA has more statistical power to identify treatment effect.
In conclusion, even though our preliminary analyses show that two treatments (warm and cool dialysis) do not have difference, they are statistically significant if we adjust the problem of chance imbalances at baseline in analysis of covariance approach. Thus, treatments have different effects on the changes of blood pressure during dialysis.
“A picture is worth a thousand words.” In our case, we are interested in whether the changes of blood pressure from the baseline to the dialysis follow-ups are different in cold and warm groups. To simplify our question, we only focus on the changes from the pre-dialysis to the first and second hours during the dialysis sessions. Also, by assuming all missing values are missing completely random, we can ignore these missing values (native method).
| temp | Mean (Systolic) | Mean (Diastolic) | Median (Systolic) | Median (Diastolic) | sd (Systolic) | sd (Diastolic) |
|---|---|---|---|---|---|---|
| hot | 15.24 | 7.31 | 15.00 | 6.00 | 18.58 | 13.00 |
| cold | 13.72 | 4.94 | 14.00 | 5.00 | 17.06 | 10.69 |
The above table summarizes the changes of blood pressure (diastolic/systolic) from the pre-dialysis to the first hour during the dialysis sessions in cold and hot (warm) groups. Overall, the median of the changes of blood pressure do not have huge differences in cold and hot (warm) groups. However, it seems that the mean and variation in cold group are smaller than those in hot (warm) group in diastolic. Therefore, two treatments (hot and cold) may not have a huge difference.
| temp | period | Mean of Difference (Systolic) | Median (Systolic) | sd (Systolic) |
|---|---|---|---|---|
| hot | 1.00 | 12.25 | 10.00 | 21.85 |
| hot | 2.00 | 16.05 | 16.00 | 17.55 |
| cold | 1.00 | 14.59 | 14.00 | 18.52 |
| cold | 2.00 | 13.36 | 14.00 | 16.42 |
| temp | period | Mean of Difference (Diastolic) | Median (Diastolic) | sd (Diastolic) |
|---|---|---|---|---|
| hot | 1.00 | 4.26 | 3.00 | 12.80 |
| hot | 2.00 | 8.13 | 7.00 | 12.94 |
| cold | 1.00 | 5.52 | 5.00 | 12.36 |
| cold | 2.00 | 4.69 | 5.00 | 9.92 |
Since we have a crossover study design, whether the participants were in warm or cold group in period 1 does matter in our analyses. The above two figures and tables summarize the changes of blood pressure (diastolic/systolic) from the pre-dialysis to the first hour during the dialysis sessions in cold and hot(warm) groups through separating participants by period. In terms of systolic, if participants were in warm group first, they would have slightly higher changes of blood pressure in cold group (period 2). However, patients would have higher changes of blood pressure in period 2 if they were in cold group in period 1. This is similar to diastolic. In this case, period may play a role in predicting the changes of blood pressure.
| temp | baseline (Systolic) | baseline (Diastolic) | Mean (Systolic 1) | Mean (Diastolic 1) | Mean (Systolic 2) | Mean (Diastolic 2) |
|---|---|---|---|---|---|---|
| hot | 145.44 | 81.42 | 130.58 | 74.20 | 129.69 | 74.31 |
| cold | 148.36 | 81.50 | 134.62 | 76.55 | 135.81 | 78.14 |
The above two figures and tables summarize the changes of blood pressure (diastolic/systolic) from the pre-dialysis to the first two hours during the dialysis sessions in cold and hot(warm) groups. It can be noticed that the blood pressure is decreasing from baseline to the first hour for all treatments in both systolic and diastolic. However, when we consider the second hour, the results have changed. The blood pressure is increasing in cold group and keeping the same in hot group (in both systolic and diastolic). Due to time limitation, we only do the confirmative analyses for the first hour. This is one limitation for this report, which can be further improved.
Preliminary Analysis
To help to confirm what we have seen in above section, we can analyze our data by following the Hill's and Armitage approach, which can be found here. In order to use this method, each participants can only have two paired measures (one for hot group and one for cold group). Thus, we took the average of the changes of blood pressure for each patients in every group. The four tables below are summarized statistics based on the Hill's and Armitage approach.
Group A (hot followed by cold) Diastolic
Period 1 Cold (Dias) Period 2 Hot (Dias) (1) - (2) ((1) + (2))/2 H - C
mean 3.96 7.79 -3.83 5.88 3.83
sd 6.83 6.83 5.82 6.18 5.82
Group B (cold followed by hot) Diastolic
Period 1 Hot (Dias) Period 2 Cold (Dias) (1) - (2) ((1) + (2))/2 H - C
mean 4.82 4.38 0.44 4.60 0.44
sd 7.60 4.11 8.21 4.53 8.21
Group A (hot followed by cold) Systolic
Period 1 Cold (Sys) Period 2 Hot (Sys) (1) - (2) ((1) + (2))/2 H - C
mean 9.76 15.82 -6.05 12.79 6.05
sd 13.22 9.99 10.99 10.35 10.99
Group B (cold followed by hot) Systolic
Period 1 Hot (Sys) Period 2 Cold (Sys) (1) - (2) ((1) + (2))/2 H - C
mean 11.31 12.99 -1.68 12.15 -1.68
sd 10.78 7.79 8.78 8.32 8.78
Based on the Hill's and Armitage's theory, we can investigate the possibility of a treatment effect, period effect, and treatment-period interaction through doing two sample t tests. The results of these t tests are summarized in the following table:
Systolic Diastolic
Treatment 0.15 0.06
Period 0.01 0.14
Treatment-Period Interaction 0.82 0.43
Above table shows that period is statistically significant in systolic, which is consistent with what we found in descriptive analysis. Thus, the corresponding p-value for treatment can be influenced by the significance of period. Therefore, in terms of treatment, the p-value in systolic may not accurate. Another interesting point is that the p-value of treatment in diastolic is just above 5% significant boundary. As I mentioned in early sections, we deleted several missing values by assuming they are missing at completely random. Therefore, the conclusion for treatment in diastolic can be easily altered if we do not have any missing values. In this situation, in order to draw more reliable conclusion, more analyses are required.
Analysis of Covariance Approach
Instead of using the Hill's and Armitage approach, we can further confirm our results by following an analysis of covariance approach (ANCOVA). You can find detailed information about ANCOVA here. Even though some people try to use change scores to fix the problem of chance imbalances at baseline between two treatment groups, analyzing change scores does not truly control for baseline imbalance problem. In particular, if participants have higher blood pressure at baseline, logically it is easier for them to decrease their blood pressure during dialysis. Therefore, ANCOVA is a better way to use. In order to use ANCOVA, we consider the baseline measurements, first hour measurements, period, temp, and the interaction of period and temp for both systolic and diastolic. We tried both linear models and mixed effect models with random effect on id. The results show that baseline, temp, and interaction are statistically significant at 5% level in both systolic and diastolic, which are totally different from what we have found in preliminary analyses. In general, compared with other methods, such as analysis of change scores, ANCOVA has more statistical power to identify treatment effect.
In conclusion, even though our preliminary analyses show that two treatments (warm and cool dialysis) do not have difference, they are statistically significant if we adjust the problem of chance imbalances at baseline in analysis of covariance approach. Thus, treatments have different effects on the changes of blood pressure during dialysis.
To help to confirm what we have seen in above section, we can analyze our data by following the Hill's and Armitage approach, which can be found here. In order to use this method, each participants can only have two paired measures (one for hot group and one for cold group). Thus, we took the average of the changes of blood pressure for each patients in every group. The four tables below are summarized statistics based on the Hill's and Armitage approach.
Group A (hot followed by cold) Diastolic
| Period 1 Cold (Dias) | Period 2 Hot (Dias) | (1) - (2) | ((1) + (2))/2 | H - C | |
|---|---|---|---|---|---|
| mean | 3.96 | 7.79 | -3.83 | 5.88 | 3.83 |
| sd | 6.83 | 6.83 | 5.82 | 6.18 | 5.82 |
Group B (cold followed by hot) Diastolic
| Period 1 Hot (Dias) | Period 2 Cold (Dias) | (1) - (2) | ((1) + (2))/2 | H - C | |
|---|---|---|---|---|---|
| mean | 4.82 | 4.38 | 0.44 | 4.60 | 0.44 |
| sd | 7.60 | 4.11 | 8.21 | 4.53 | 8.21 |
Group A (hot followed by cold) Systolic
| Period 1 Cold (Sys) | Period 2 Hot (Sys) | (1) - (2) | ((1) + (2))/2 | H - C | |
|---|---|---|---|---|---|
| mean | 9.76 | 15.82 | -6.05 | 12.79 | 6.05 |
| sd | 13.22 | 9.99 | 10.99 | 10.35 | 10.99 |
Group B (cold followed by hot) Systolic
| Period 1 Hot (Sys) | Period 2 Cold (Sys) | (1) - (2) | ((1) + (2))/2 | H - C | |
|---|---|---|---|---|---|
| mean | 11.31 | 12.99 | -1.68 | 12.15 | -1.68 |
| sd | 10.78 | 7.79 | 8.78 | 8.32 | 8.78 |
Based on the Hill's and Armitage's theory, we can investigate the possibility of a treatment effect, period effect, and treatment-period interaction through doing two sample t tests. The results of these t tests are summarized in the following table:
| Systolic | Diastolic | |
|---|---|---|
| Treatment | 0.15 | 0.06 |
| Period | 0.01 | 0.14 |
| Treatment-Period Interaction | 0.82 | 0.43 |
Above table shows that period is statistically significant in systolic, which is consistent with what we found in descriptive analysis. Thus, the corresponding p-value for treatment can be influenced by the significance of period. Therefore, in terms of treatment, the p-value in systolic may not accurate. Another interesting point is that the p-value of treatment in diastolic is just above 5% significant boundary. As I mentioned in early sections, we deleted several missing values by assuming they are missing at completely random. Therefore, the conclusion for treatment in diastolic can be easily altered if we do not have any missing values. In this situation, in order to draw more reliable conclusion, more analyses are required.
Analysis of Covariance Approach
Instead of using the Hill's and Armitage approach, we can further confirm our results by following an analysis of covariance approach (ANCOVA). You can find detailed information about ANCOVA here. Even though some people try to use change scores to fix the problem of chance imbalances at baseline between two treatment groups, analyzing change scores does not truly control for baseline imbalance problem. In particular, if participants have higher blood pressure at baseline, logically it is easier for them to decrease their blood pressure during dialysis. Therefore, ANCOVA is a better way to use. In order to use ANCOVA, we consider the baseline measurements, first hour measurements, period, temp, and the interaction of period and temp for both systolic and diastolic. We tried both linear models and mixed effect models with random effect on id. The results show that baseline, temp, and interaction are statistically significant at 5% level in both systolic and diastolic, which are totally different from what we have found in preliminary analyses. In general, compared with other methods, such as analysis of change scores, ANCOVA has more statistical power to identify treatment effect.
In conclusion, even though our preliminary analyses show that two treatments (warm and cool dialysis) do not have difference, they are statistically significant if we adjust the problem of chance imbalances at baseline in analysis of covariance approach. Thus, treatments have different effects on the changes of blood pressure during dialysis.
Instead of using the Hill's and Armitage approach, we can further confirm our results by following an analysis of covariance approach (ANCOVA). You can find detailed information about ANCOVA here. Even though some people try to use change scores to fix the problem of chance imbalances at baseline between two treatment groups, analyzing change scores does not truly control for baseline imbalance problem. In particular, if participants have higher blood pressure at baseline, logically it is easier for them to decrease their blood pressure during dialysis. Therefore, ANCOVA is a better way to use. In order to use ANCOVA, we consider the baseline measurements, first hour measurements, period, temp, and the interaction of period and temp for both systolic and diastolic. We tried both linear models and mixed effect models with random effect on id. The results show that baseline, temp, and interaction are statistically significant at 5% level in both systolic and diastolic, which are totally different from what we have found in preliminary analyses. In general, compared with other methods, such as analysis of change scores, ANCOVA has more statistical power to identify treatment effect.
In conclusion, even though our preliminary analyses show that two treatments (warm and cool dialysis) do not have difference, they are statistically significant if we adjust the problem of chance imbalances at baseline in analysis of covariance approach. Thus, treatments have different effects on the changes of blood pressure during dialysis.