What determines excrement's caliber?
With the help of Bristol Stool Chart (BSC) and full explanation of conditions related to different stool sizes from this link you could have a full understanding over stool size.




  » Type 1: Separate hard lumps, like nuts Typical for acute
  disbacteriosis. These stools lack a normal amorphous quality, because
  bacteria are missing and there is nothing to retain water. The lumps
  are hard and abrasive, the typical diameter ranges from 1 to 2 cm
  (0.4–0.8”), and they're painful to pass, because the lumps are hard
  and scratchy. There is a high likelihood of anorectal bleeding from
  mechanical laceration of the anal canal. Typical for post-antibiotic
  treatments and for people attempting fiber-free (low-carb) diets.
  Flatulence isn't likely, because fermentation of fiber isn't taking
  place.
  
  » Type 2: Sausage-like but lumpy Represents a combination of Type 1
  stools impacted into a single mass and lumped together by fiber
  components and some bacteria. Typical for organic constipation. The
  diameter is 3 to 4 cm (1.2–1.6”). This type is the most destructive by
  far because its size is near or exceeds the maximum opening of the
  anal canal‘s aperture (3.5 cm). It‘s bound to cause extreme straining
  during elimination, and most likely to cause anal canal laceration,
  hemorrhoidal prolapse, or diverticulosis. To attain this form, the
  stools must be in the colon for at least several weeks instead of the
  normal 72 hours. Anorectal pain, hemorrhoidal disease, anal fissures,
  withholding or delaying of defecation, and a history of chronic
  constipation are the most likely causes. Minor flatulence is probable.
  A person experiencing these stools is most likely to suffer from
  irritable bowel syndrome because of continuous pressure of large
  stools on the intestinal walls. The possibility of obstruction of the
  small intestine is high, because the large intestine is filled to
  capacity with stools. Adding supplemental fiber to expel these stools
  is dangerous, because the expanded fiber has no place to go, and may
  cause hernia, obstruction, or perforation of the small and large
  intestine alike.
  
  » Type 3: Like a sausage but with cracks in the surface This form has
  all of the characteristics of Type 2 stools, but the transit time is
  faster, between one and two weeks. Typical for latent constipation.
  The diameter is 2 to 3.5 cm (0.8–1.4”). Irritable bowel syndrome is
  likely. Flatulence is minor, because of disbacteriosis. The fact that
  it hasn't became as enlarged as Type 2 suggests that the defecations
  are regular. Straining is required. All of the adverse effects typical
  for Type 2 stools are likely for type 3, especially the rapid
  deterioration of hemorrhoidal disease.
  
  » Type 4: Like a sausage or snake, smooth and soft This form is normal
  for someone defecating once daily. The diameter is 1 to 2 cm
  (0.4–0.8”). The larger diameter suggests a longer transit time or a
  large amount of dietary fiber in the diet.
  
  » Type 5: Soft blobs with clear-cut edges I consider this form ideal.
  It is typical for a person who has stools twice or three times daily,
  after major meals. The diameter is 1 to 1.5 cm (0.4–0.6”).
  
  » Type 6: Fluffy pieces with ragged edges, a mushy stool This form is
  close to the margins of comfort in several respects. First, it may be
  difficult to control the urge, especially when you don‘t have
  immediate access to a bathroom. Second, it is a rather messy affair to
  manage with toilet paper alone, unless you have access to a flexible
  shower or bidet. Otherwise, I consider it borderline normal. These
  kind of stools may suggest a slightly hyperactive colon (fast
  motility), excess dietary potassium, or sudden dehydration or spike in
  blood pressure related to stress (both cause the rapid release of
  water and potassium from blood plasma into the intestinal cavity). It
  can also indicate a hypersensitive personality prone to stress, too
  many spices, drinking water with a high mineral content, or the use of
  osmotic (mineral salts) laxatives.
  
  » Type 7: Watery, no solid pieces This, of course, is diarrhea, a
  subject outside the scope of this chapter with just one important and
  notable exception—so-called paradoxical diarrhea. It‘s typical for
  people (especially young children and infirm or convalescing adults)
  affected by fecal impaction—a condition that follows or accompanies
  type 1 stools. During paradoxical diarrhea the liquid contents of the
  small intestine (up to 1.5–2 liters/quarts daily) have no place to go
  but down, because the large intestine is stuffed with impacted stools
  throughout its entire length. Some water gets absorbed, the rest
  accumulates in the rectum. The reason this type of diarrhea is called
  paradoxical is not because its nature isn‘t known or understood, but
  because being severely constipated and experiencing diarrhea all at
  once, is, indeed, a paradoxical situation. Unfortunately, it‘s all too
  common.
  
  Interestingly, the interpretations and explanations of the BSF scale
  that accompany the original chart differ from my analysis. To this I
  can only say: thanks for great pictures, but, no thanks for the
  rest...
  
  Excerpted from Fiber Menace, page 117-120; BSF Chart: wikipedia.org


But to make it more clear:

Reasons behind thicker and harder stool:


Lower fiber food
Lower liquid intake
Lower physical activity
Lack or lower amount of bacteria related to retaining water inside rectum and colon
Medication:    


  Many medications have constipation as a side effect. Some include (but
  are not limited to) opioids also known as narcotics, diuretics,
  antidepressants, antihistamines, antispasmodics, anticonvulsants, and
  aluminum antacids. Certain calcium channel blockers such as nifedipine
  and verapamil can cause severe constipation due to dysfunction of
  motility in the rectosigmoid colon.

Metabolic and muscular


  Metabolic and endocrine problems which may lead to constipation
  include: hypercalcemia, hypothyroidism, diabetes mellitus, cystic
  fibrosis, and celiac disease. Constipation is also common in
  individuals with muscular and myotonic dystrophy.

Structural and functional abnormalities

Any physical digestive tract line problem which cause interference of stool movement. Like cancer or some kind of inflammation that could ban the canal


  Constipation has a number of structural (mechanical, morphological,
  anatomical) causes, including: spinal cord lesions, Parkinson, colon
  cancer, anal fissures, proctitis, and pelvic floor dysfunction.
  
  Constipation also has functional (neurological) causes, including anismus, descending perineum syndrome, and Hirschsprung's disease. In
  infants, Hirschsprung's disease is the most common medical disorder
  associated with constipation. Anismus occurs in a small minority of
  persons with chronic constipation or obstructed defecation.

Psychological


  Voluntary withholding of the stool is a common cause of
  constipation.The choice to withhold can be due to factors such as fear
  of pain, fear of public restrooms, or laziness. When a child holds in
  the stool a combination of encouragement, fluids, fiber, and laxatives
  may be useful to overcome the problem.



Edit:

In the second provided link you could get how cross-sectional diameter changes based on what you eat or how you behave with your body mentally or physically if you are a healthy person.

But as you are interested more on cross-sectional size of stool and reason behind it, consider the fact that colons and rectum are water absorber. The more stool stays there the more thick and dry it gets. So if you ate more fiber you will probably have softer stool and thinner caliber as the result of pressure of the rectum canal. Because some bacteria which are active in the fibers of stool are water absorbers too and the reason behind why doctors insist on eating more probiotic products is to keep these bacteria present and stable and also to get vitamins from their activities. So you could change your caliber only by eating more fiber more water and having more physical activities and also for consistent thick and dry stool (constipation) individual must share the problem with doctor because it could bring more dangerous problems in future or maybe it's a sign of some other serious disease that person's body is already dealing with. Most of the time it's nothing but our diet and hours we are sitting for variety of reasons or maybe focusing on another activitys (work, computer games, disabilities, watching tv and so on and so forth).

https://en.wikipedia.org/wiki/Constipation
https://www.gutsense.org/constipation/normal_stools.html