Interpreting a p-value of 1 from a 1-sample t-test

Question

I have this data frame (reproducible):

structure(list(age = c(62.84998, 60.33899, 52.74698, 42.38498, 
 79.88495, 93.01599, 62.37097, 86.83899, 85.65594, 42.25897), 
     death = c(0, 1, 1, 1, 0, 1, 1, 1, 1, 1), sex = c("male", 
     "female", "female", "female", "female", "male", "male", "male", 
     "male", "female"), hospdead = c(0, 1, 0, 0, 0, 1, 0, 0, 0, 
     0), slos = c(5, 4, 17, 3, 16, 4, 9, 7, 12, 8), d.time = c(2029, 
     4, 47, 133, 2029, 4, 659, 142, 63, 370), dzgroup = c("Lung Cancer", 
     "Cirrhosis", "Cirrhosis", "Lung Cancer", "ARF/MOSF w/Sepsis", 
     "Coma", "CHF", "CHF", "Lung Cancer", "Colon Cancer"), dzclass = c("Cancer", 
     "COPD/CHF/Cirrhosis", "COPD/CHF/Cirrhosis", "Cancer", "ARF/MOSF", 
     "Coma", "COPD/CHF/Cirrhosis", "COPD/CHF/Cirrhosis", "Cancer", 
     "Cancer"), num.co = c(0, 2, 2, 2, 1, 1, 1, 3, 2, 0), edu = c(11, 
     12, 12, 11, NA, 14, 14, NA, 12, 11), income = c("$11-$25k", 
     "$11-$25k", "under $11k", "under $11k", NA, NA, "$25-$50k", 
     NA, NA, "$25-$50k"), scoma = c(0, 44, 0, 0, 26, 55, 0, 26, 
     26, 0), charges = c(9715, 34496, 41094, 3075, 50127, 6884, 
     30460, 30460, NA, 9914), totcst = c(NA_real_, NA_real_, NA_real_, 
     NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, 
     NA_real_), totmcst = c(NA_real_, NA_real_, NA_real_, NA_real_, 
     NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, NA_real_
     ), avtisst = c(7, 29, 13, 7, 18.666656, 5, 8, 6.5, 8.5, 8
     ), race = c("other", "white", "white", "white", "white", 
     "white", "white", "white", "black", "hispanic"), sps = c(33.8984375, 
     52.6953125, 20.5, 20.0976562, 23.5, 19.3984375, 17.296875, 
     21.5976562, 15.8984375, 2.2998047), aps = c(20, 74, 45, 19, 
     30, 27, 46, 53, 17, 9), surv2m = c(0.262939453, 0.0009999275, 
     0.790893555, 0.698974609, 0.634887695, 0.284973145, 0.892944336, 
     0.670898438, 0.570922852, 0.952880859), surv6m = c(0.0369949341, 
     0, 0.664916992, 0.411987305, 0.532958984, 0.214996338, 0.820922852, 
     0.498962402, 0.24899292, 0.887939453), hday = c(1, 3, 4, 
     1, 3, 1, 1, 1, 1, 1), diabetes = c(0, 0, 0, 0, 0, 0, 0, 1, 
     0, 0), dementia = c(0, 0, 0, 0, 0, 0, 0, 0, 1, 0), ca = c("metastatic", 
     "no", "no", "metastatic", "no", "no", "no", "no", "metastatic", 
     "metastatic"), prg2m = c(0.5, 0, 0.75, 0.899999619, 0.899999619, 
     0, NA, 0.799999714, 0.049999982, NA), prg6m = c(0.25, 0, 
     0.5, 0.5, 0.8999996, 0, 0.6999998, 0.3999999, 0.0001249999, 
     NA), dnr = c("no dnr", NA, "no dnr", "no dnr", "no dnr", 
     "no dnr", "no dnr", "no dnr", "dnr after sadm", "no dnr"), 
     dnrday = c(5, NA, 17, 3, 16, 4, 9, 7, 2, 8), meanbp = c(97, 
     43, 70, 75, 59, 110, 78, 72, 97, 84), wblc = c(6, 17.0976562, 
     8.5, 9.09960938, 13.5, 10.3984375, 11.6992188, 13.5996094, 
     9.69921875, 11.2988281), hrt = c(69, 112, 88, 88, 112, 101, 
     120, 100, 56, 94), resp = c(22, 34, 28, 32, 20, 44, 28, 26, 
     20, 20), temp = c(36, 34.59375, 37.39844, 35, 37.89844, 38.39844, 
     37.39844, 37.59375, 36.59375, 38.19531), pafi = c(388, 98, 
     231.65625, NA, 173.3125, 266.625, 309.5, 404.75, 357.125, 
     NA), alb = c(1.7998047, NA, NA, NA, NA, NA, 4.7998047, NA, 
     NA, 4.6992188), bili = c(0.19998169, NA, 2.19970703, NA, 
     NA, NA, 0.39996338, NA, 0.39996338, 0.19998169), crea = c(1.19995117, 
     5.5, 2, 0.79992676, 0.79992676, 0.69995117, 1.59985352, 2, 
     1, 0.79992676), sod = c(141, 132, 134, 139, 143, 140, 132, 
     139, 143, 139), ph = c(7.459961, 7.25, 7.459961, NA, 7.509766, 
     7.65918, 7.479492, 7.509766, 7.449219, NA), glucose = c(NA_real_, 
     NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, 
     NA_real_, NA_real_, NA_real_), bun = c(NA_real_, NA_real_, 
     NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, 
     NA_real_, NA_real_), urine = c(NA_real_, NA_real_, NA_real_, 
     NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, NA_real_, 
     NA_real_), adlp = c(7, NA, 1, 0, NA, NA, 0, NA, NA, 0), adls = c(7, 
     1, 0, 0, 2, 1, 1, 0, 7, NA), sfdm2 = c(NA, "<2 mo. follow-up", 
     "<2 mo. follow-up", "no(M2 and SIP pres)", "no(M2 and SIP pres)", 
     "<2 mo. follow-up", "no(M2 and SIP pres)", NA, NA, NA), adlsc = c(7, 
     1, 0, 0, 2, 1, 1, 0, 7, 0.4947999)), row.names = c(NA, 10L
 ), class = "data.frame")

I am wanting to estimate the population mean serum Sodium level (the variable is labeled as sod) on the patient’s third study day and determine if there evidence to support the claim that the population mean serum Sodium level on the patient’s third study day is above 142. I'd also like to get the 95% confidence intervals. This is what I have done.

SB_xlsx9 = read.xlsx("~/Desktop/support2.xlsx", sheet=2, colNames=TRUE)
SB_xlsx9 = SB_xlsx9[!is.na(SB_xlsx9$sod), ]

This creates my dataset and removes all NAs from the sod column.

I then performed a 1-sample t-test as shown below with output.

alpha  = 0.05
x_bar  = mean(SB_xlsx9$sod)
mu_0   = 142
n      = length(SB_xlsx9$sod)
t.test(x=SB_xlsx9$sod, mu=mu_0, n=n, alternative='greater')
## 
##  One Sample t-test
## 
## data:  SB_xlsx9$sod
## t = -70.128, df = 9103, p-value = 1
## alternative hypothesis: true mean is greater than 142
## 95 percent confidence interval:
##  137.4646      Inf
## sample estimates:
## mean of x 
##  137.5685

I want to make sure my interpretation here is correct. The estimate population mean of serum Sodium level on day 3 is 137.6 (rounded). Based on this result, I would fail to reject the null hypothesis that the population mean is not greater than 142. I guess my main question though, is having a p-value of 1 okay? I'm a little confused if I have done something wrong for my t-test, since I have never had a result with a p-value of 1. The reproducible data is only a sample of the top 10 rows, but I just feel like something isn't right here. UPDATE. I added a photo of the histogram from my full dataset, so I think the p-value of 1 does make a bit more sense.

Answer 1

Looks like you are examining the statistical results without examining the raw values closely enough. From the 10 data points given, your sample mean is 138.2 with a standard error of about 1.3. That is, your sample estimate of the population mean is almost three standard errors below the null hypothesised value of 142. If those ten data points are representative of the whole 9 thousand, you should not be surprised that the p-value for the test of whether the sample based estimate is greater than 142 comes out close to 1.

Plot the data values as a histogram to see how they sit relative to the 142 null hypothesis and your t-test result might make more sense.